Theory AOT_PLM
1
2theory AOT_PLM
3 imports AOT_axioms
4begin
5
6
7section‹The Deductive System PLM›
8
9
10unbundle AOT_no_atp
11
12
13
14
15interpretation AOT_no_meta_syntax.
16
17
18unbundle AOT_syntax
19
20
21
22AOT_theorem "modus-ponens": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
23 using assms by (simp add: AOT_sem_imp)
24lemmas MP = "modus-ponens"
25
26AOT_theorem "non-con-thm-thm": assumes ‹❙⊢⇩□ φ› shows ‹❙⊢ φ›
27 using assms by simp
28
29AOT_theorem "vdash-properties:1[1]": assumes ‹φ ∈ Λ› shows ‹❙⊢ φ›
30 using assms unfolding AOT_model_act_axiom_def by blast
31
32attribute_setup act_axiom_inst =
33 ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))›
34 "Instantiate modally fragile axiom as modally fragile theorem."
35
36AOT_theorem "vdash-properties:1[2]": assumes ‹φ ∈ Λ⇩□› shows ‹❙⊢⇩□ φ›
37 using assms unfolding AOT_model_axiom_def by blast
38
39attribute_setup axiom_inst =
40 ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[2]"})))›
41 "Instantiate axiom as theorem."
42
43method cqt_2_lambda_inst_prover = (fast intro: AOT_instance_of_cqt_2_intro)
44method "cqt:2[lambda]" = (rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
45lemmas "cqt:2" = "cqt:2[const_var]"[axiom_inst] "cqt:2[lambda]"[axiom_inst] AOT_instance_of_cqt_2_intro
46method "cqt:2" = (safe intro!: "cqt:2")
47
48AOT_theorem "vdash-properties:3": assumes ‹❙⊢⇩□ φ› shows ‹Γ ❙⊢ φ›
49 using assms by blast
50
51AOT_theorem "vdash-properties:5": assumes ‹Γ⇩1 ❙⊢ φ› and ‹Γ⇩2 ❙⊢ φ → ψ› shows ‹Γ⇩1, Γ⇩2 ❙⊢ ψ›
52 using MP assms by blast
53
54AOT_theorem "vdash-properties:6": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
55 using MP assms by blast
56
57AOT_theorem "vdash-properties:8": assumes ‹Γ ❙⊢ φ› and ‹φ ❙⊢ ψ› shows ‹Γ ❙⊢ ψ›
58 using assms by argo
59
60AOT_theorem "vdash-properties:9": assumes ‹φ› shows ‹ψ → φ›
61 using MP "pl:1"[axiom_inst] assms by blast
62
63AOT_theorem "vdash-properties:10": assumes ‹φ → ψ› and ‹φ› shows ‹ψ›
64 using MP assms by blast
65lemmas "→E" = "vdash-properties:10"
66
67AOT_theorem "rule-gen": assumes ‹for arbitrary α: φ{α}› shows ‹∀α φ{α}›
68 using assms by (metis AOT_var_of_term_inverse AOT_sem_denotes AOT_sem_forall)
69lemmas GEN = "rule-gen"
70
71AOT_theorem "RN[prem]": assumes ‹Γ ❙⊢⇩□ φ› shows ‹□Γ ❙⊢⇩□ □φ›
72 by (meson AOT_sem_box assms image_iff)
73AOT_theorem RN: assumes ‹❙⊢⇩□ φ› shows ‹□φ›
74 using "RN[prem]" assms by blast
75
76AOT_axiom "df-rules-formulas[1]": assumes ‹φ ≡⇩d⇩f ψ› shows ‹φ → ψ›
77 using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp)
78AOT_axiom "df-rules-formulas[2]": assumes ‹φ ≡⇩d⇩f ψ› shows ‹ψ → φ›
79 using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp)
80
81AOT_theorem "df-rules-formulas[3]": assumes ‹φ ≡⇩d⇩f ψ› shows ‹φ → ψ›
82 using "df-rules-formulas[1]"[axiom_inst, OF assms].
83AOT_theorem "df-rules-formulas[4]": assumes ‹φ ≡⇩d⇩f ψ› shows ‹ψ → φ›
84 using "df-rules-formulas[2]"[axiom_inst, OF assms].
85
86
87AOT_axiom "df-rules-terms[1]":
88 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}›
89 shows ‹(σ{τ⇩1...τ⇩n}↓ → τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}) & (¬σ{τ⇩1...τ⇩n}↓ → ¬τ{τ⇩1...τ⇩n}↓)›
90 using assms by (simp add: AOT_model_axiomI AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def)
91AOT_axiom "df-rules-terms[2]":
92 assumes ‹τ =⇩d⇩f σ›
93 shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
94 by (metis "df-rules-terms[1]" case_unit_Unity assms)
95
96AOT_theorem "df-rules-terms[3]":
97 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}›
98 shows ‹(σ{τ⇩1...τ⇩n}↓ → τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}) & (¬σ{τ⇩1...τ⇩n}↓ → ¬τ{τ⇩1...τ⇩n}↓)›
99 using "df-rules-terms[1]"[axiom_inst, OF assms].
100AOT_theorem "df-rules-terms[4]":
101 assumes ‹τ =⇩d⇩f σ›
102 shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
103 using "df-rules-terms[2]"[axiom_inst, OF assms].
104
105
106AOT_theorem "if-p-then-p": ‹φ → φ›
107 by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)
108
109AOT_theorem "deduction-theorem": assumes ‹φ ❙⊢ ψ› shows ‹φ → ψ›
110 using assms by (simp add: AOT_sem_imp)
111lemmas CP = "deduction-theorem"
112lemmas "→I" = "deduction-theorem"
113
114AOT_theorem "ded-thm-cor:1": assumes ‹Γ⇩1 ❙⊢ φ → ψ› and ‹Γ⇩2 ❙⊢ ψ → χ› shows ‹Γ⇩1, Γ⇩2 ❙⊢ φ → χ›
115 using "→E" "→I" assms by blast
116AOT_theorem "ded-thm-cor:2": assumes ‹Γ⇩1 ❙⊢ φ → (ψ → χ)› and ‹Γ⇩2 ❙⊢ ψ› shows ‹Γ⇩1, Γ⇩2 ❙⊢ φ → χ›
117 using "→E" "→I" assms by blast
118
119AOT_theorem "ded-thm-cor:3": assumes ‹φ → ψ› and ‹ψ → χ› shows ‹φ → χ›
120 using "→E" "→I" assms by blast
121declare "ded-thm-cor:3"[trans]
122AOT_theorem "ded-thm-cor:4": assumes ‹φ → (ψ → χ)› and ‹ψ› shows ‹φ → χ›
123 using "→E" "→I" assms by blast
124
125lemmas "Hypothetical Syllogism" = "ded-thm-cor:3"
126
127AOT_theorem "useful-tautologies:1": ‹¬¬φ → φ›
128 by (metis "pl:3"[axiom_inst] "→I" "Hypothetical Syllogism")
129AOT_theorem "useful-tautologies:2": ‹φ → ¬¬φ›
130 by (metis "pl:3"[axiom_inst] "→I" "ded-thm-cor:4")
131AOT_theorem "useful-tautologies:3": ‹¬φ → (φ → ψ)›
132 by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "→I")
133AOT_theorem "useful-tautologies:4": ‹(¬ψ → ¬φ) → (φ → ψ)›
134 by (meson "pl:3"[axiom_inst] "Hypothetical Syllogism" "→I")
135AOT_theorem "useful-tautologies:5": ‹(φ → ψ) → (¬ψ → ¬φ)›
136 by (metis "useful-tautologies:4" "Hypothetical Syllogism" "→I")
137
138AOT_theorem "useful-tautologies:6": ‹(φ → ¬ψ) → (ψ → ¬φ)›
139 by (metis "→I" MP "useful-tautologies:4")
140
141AOT_theorem "useful-tautologies:7": ‹(¬φ → ψ) → (¬ψ → φ)›
142 by (metis "→I" MP "useful-tautologies:3" "useful-tautologies:5")
143
144AOT_theorem "useful-tautologies:8": ‹φ → (¬ψ → ¬(φ → ψ))›
145 by (metis "→I" MP "useful-tautologies:5")
146
147AOT_theorem "useful-tautologies:9": ‹(φ → ψ) → ((¬φ → ψ) → ψ)›
148 by (metis "→I" MP "useful-tautologies:6")
149
150AOT_theorem "useful-tautologies:10": ‹(φ → ¬ψ) → ((φ → ψ) → ¬φ)›
151 by (metis "→I" MP "pl:3"[axiom_inst])
152
153AOT_theorem "dn-i-e:1": assumes ‹φ› shows ‹¬¬φ›
154 using MP "useful-tautologies:2" assms by blast
155lemmas "¬¬I" = "dn-i-e:1"
156AOT_theorem "dn-i-e:2": assumes ‹¬¬φ› shows ‹φ›
157 using MP "useful-tautologies:1" assms by blast
158lemmas "¬¬E" = "dn-i-e:2"
159
160AOT_theorem "modus-tollens:1": assumes ‹φ → ψ› and ‹¬ψ› shows ‹¬φ›
161 using MP "useful-tautologies:5" assms by blast
162AOT_theorem "modus-tollens:2": assumes ‹φ → ¬ψ› and ‹ψ› shows ‹¬φ›
163 using "¬¬I" "modus-tollens:1" assms by blast
164lemmas MT = "modus-tollens:1" "modus-tollens:2"
165
166AOT_theorem "contraposition:1[1]": assumes ‹φ → ψ› shows ‹¬ψ → ¬φ›
167 using "→I" MT(1) assms by blast
168AOT_theorem "contraposition:1[2]": assumes ‹¬ψ → ¬φ› shows ‹φ → ψ›
169 using "→I" "¬¬E" MT(2) assms by blast
170
171AOT_theorem "contraposition:2": assumes ‹φ → ¬ψ› shows ‹ψ → ¬φ›
172 using "→I" MT(2) assms by blast
173
174
175AOT_theorem "reductio-aa:1":
176 assumes ‹¬φ ❙⊢ ¬ψ› and ‹¬φ ❙⊢ ψ› shows ‹φ›
177 using "→I" "¬¬E" MT(2) assms by blast
178AOT_theorem "reductio-aa:2":
179 assumes ‹φ ❙⊢ ¬ψ› and ‹φ ❙⊢ ψ› shows ‹¬φ›
180 using "reductio-aa:1" assms by blast
181lemmas "RAA" = "reductio-aa:1" "reductio-aa:2"
182
183AOT_theorem "exc-mid": ‹φ ∨ ¬φ›
184 using "df-rules-formulas[4]" "if-p-then-p" MP "conventions:2" by blast
185
186AOT_theorem "non-contradiction": ‹¬(φ & ¬φ)›
187 using "df-rules-formulas[3]" MT(2) "useful-tautologies:2" "conventions:1" by blast
188
189AOT_theorem "con-dis-taut:1": ‹(φ & ψ) → φ›
190 by (meson "→I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
191AOT_theorem "con-dis-taut:2": ‹(φ & ψ) → ψ›
192 by (metis "→I" "df-rules-formulas[3]" MT(2) RAA(2) "¬¬E" "conventions:1")
193lemmas "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"
194
195AOT_theorem "con-dis-taut:3": ‹φ → (φ ∨ ψ)›
196 by (meson "contraposition:1[2]" "df-rules-formulas[4]" MP "→I" "conventions:2")
197AOT_theorem "con-dis-taut:4": ‹ψ → (φ ∨ ψ)›
198 using "Hypothetical Syllogism" "df-rules-formulas[4]" "pl:1"[axiom_inst] "conventions:2" by blast
199lemmas "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"
200
201AOT_theorem "con-dis-taut:5": ‹φ → (ψ → (φ & ψ))›
202 by (metis "contraposition:2" "Hypothetical Syllogism" "→I" "df-rules-formulas[4]" "conventions:1")
203lemmas Adjunction = "con-dis-taut:5"
204
205AOT_theorem "con-dis-taut:6": ‹(φ & φ) ≡ φ›
206 by (metis Adjunction "→I" "df-rules-formulas[4]" MP "Conjunction Simplification"(1) "conventions:3")
207lemmas "Idempotence of &" = "con-dis-taut:6"
208
209AOT_theorem "con-dis-taut:7": ‹(φ ∨ φ) ≡ φ›
210proof -
211 {
212 AOT_assume ‹φ ∨ φ›
213 AOT_hence ‹¬φ → φ›
214 using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
215 AOT_hence ‹φ› using "if-p-then-p" RAA(1) MP by blast
216 }
217 moreover {
218 AOT_assume ‹φ›
219 AOT_hence ‹φ ∨ φ› using "Disjunction Addition"(1) MP by blast
220 }
221 ultimately AOT_show ‹(φ ∨ φ) ≡ φ›
222 using "conventions:3"[THEN "df-rules-formulas[4]"] MP
223 by (metis Adjunction "→I")
224qed
225lemmas "Idempotence of ∨" = "con-dis-taut:7"
226
227
228AOT_theorem "con-dis-i-e:1": assumes ‹φ› and ‹ψ› shows ‹φ & ψ›
229 using Adjunction MP assms by blast
230lemmas "&I" = "con-dis-i-e:1"
231
232AOT_theorem "con-dis-i-e:2:a": assumes ‹φ & ψ› shows ‹φ›
233 using "Conjunction Simplification"(1) MP assms by blast
234AOT_theorem "con-dis-i-e:2:b": assumes ‹φ & ψ› shows ‹ψ›
235 using "Conjunction Simplification"(2) MP assms by blast
236lemmas "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"
237
238AOT_theorem "con-dis-i-e:3:a": assumes ‹φ› shows ‹φ ∨ ψ›
239 using "Disjunction Addition"(1) MP assms by blast
240AOT_theorem "con-dis-i-e:3:b": assumes ‹ψ› shows ‹φ ∨ ψ›
241 using "Disjunction Addition"(2) MP assms by blast
242AOT_theorem "con-dis-i-e:3:c": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → Θ› shows ‹χ ∨ Θ›
243 by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2) "df-rules-formulas[3]" MT(1) RAA(1) "conventions:2" assms)
244lemmas "∨I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"
245
246AOT_theorem "con-dis-i-e:4:a": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → χ› shows ‹χ›
247 by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
248AOT_theorem "con-dis-i-e:4:b": assumes ‹φ ∨ ψ› and ‹¬φ› shows ‹ψ›
249 using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
250AOT_theorem "con-dis-i-e:4:c": assumes ‹φ ∨ ψ› and ‹¬ψ› shows ‹φ›
251 using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
252lemmas "∨E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"
253
254AOT_theorem "raa-cor:1": assumes ‹¬φ ❙⊢ ψ & ¬ψ› shows ‹φ›
255 using "&E" "∨E"(3) "∨I"(2) RAA(2) assms by blast
256AOT_theorem "raa-cor:2": assumes ‹φ ❙⊢ ψ & ¬ψ› shows ‹¬φ›
257 using "raa-cor:1" assms by blast
258AOT_theorem "raa-cor:3": assumes ‹φ› and ‹¬ψ ❙⊢ ¬φ› shows ‹ψ›
259 using RAA assms by blast
260AOT_theorem "raa-cor:4": assumes ‹¬φ› and ‹¬ψ ❙⊢ φ› shows ‹ψ›
261 using RAA assms by blast
262AOT_theorem "raa-cor:5": assumes ‹φ› and ‹ψ ❙⊢ ¬φ› shows ‹¬ψ›
263 using RAA assms by blast
264AOT_theorem "raa-cor:6": assumes ‹¬φ› and ‹ψ ❙⊢ φ› shows ‹¬ψ›
265 using RAA assms by blast
266
267
268AOT_theorem "oth-class-taut:1:a": ‹(φ → ψ) ≡ ¬(φ & ¬ψ)›
269 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
270 (metis "&E" "&I" "raa-cor:3" "→I" MP)
271AOT_theorem "oth-class-taut:1:b": ‹¬(φ → ψ) ≡ (φ & ¬ψ)›
272 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
273 (metis "&E" "&I" "raa-cor:3" "→I" MP)
274AOT_theorem "oth-class-taut:1:c": ‹(φ → ψ) ≡ (¬φ ∨ ψ)›
275 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
276 (metis "&I" "∨I"(1, 2) "∨E"(3) "→I" MP "raa-cor:1")
277
278AOT_theorem "oth-class-taut:2:a": ‹(φ & ψ) ≡ (ψ & φ)›
279 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
280 (meson "&I" "&E" "→I")
281lemmas "Commutativity of &" = "oth-class-taut:2:a"
282AOT_theorem "oth-class-taut:2:b": ‹(φ & (ψ & χ)) ≡ ((φ & ψ) & χ)›
283 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
284 (metis "&I" "&E" "→I")
285lemmas "Associativity of &" = "oth-class-taut:2:b"
286AOT_theorem "oth-class-taut:2:c": ‹(φ ∨ ψ) ≡ (ψ ∨ φ)›
287 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
288 (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
289lemmas "Commutativity of ∨" = "oth-class-taut:2:c"
290AOT_theorem "oth-class-taut:2:d": ‹(φ ∨ (ψ ∨ χ)) ≡ ((φ ∨ ψ) ∨ χ)›
291 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
292 (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
293lemmas "Associativity of ∨" = "oth-class-taut:2:d"
294AOT_theorem "oth-class-taut:2:e": ‹(φ ≡ ψ) ≡ (ψ ≡ φ)›
295 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I";
296 metis "&I" "df-rules-formulas[4]" "conventions:3" "&E" "Hypothetical Syllogism" "→I" "df-rules-formulas[3]")
297lemmas "Commutativity of ≡" = "oth-class-taut:2:e"
298AOT_theorem "oth-class-taut:2:f": ‹(φ ≡ (ψ ≡ χ)) ≡ ((φ ≡ ψ) ≡ χ)›
299 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
300 "→I" "→E" "&E" "&I"
301 by metis
302lemmas "Associativity of ≡" = "oth-class-taut:2:f"
303
304AOT_theorem "oth-class-taut:3:a": ‹φ ≡ φ›
305 using "&I" "vdash-properties:6" "if-p-then-p" "df-rules-formulas[4]" "conventions:3" by blast
306AOT_theorem "oth-class-taut:3:b": ‹φ ≡ ¬¬φ›
307 using "&I" "useful-tautologies:1" "useful-tautologies:2" "vdash-properties:6" "df-rules-formulas[4]" "conventions:3" by blast
308AOT_theorem "oth-class-taut:3:c": ‹¬(φ ≡ ¬φ)›
309 by (metis "&E" "→E" RAA "df-rules-formulas[3]" "conventions:3")
310
311AOT_theorem "oth-class-taut:4:a": ‹(φ → ψ) → ((ψ → χ) → (φ → χ))›
312 by (metis "→E" "→I")
313AOT_theorem "oth-class-taut:4:b": ‹(φ ≡ ψ) ≡ (¬φ ≡ ¬ψ)›
314 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
315 "→I" "→E" "&E" "&I" RAA by metis
316AOT_theorem "oth-class-taut:4:c": ‹(φ ≡ ψ) → ((φ → χ) ≡ (ψ → χ))›
317 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
318 "→I" "→E" "&E" "&I" by metis
319AOT_theorem "oth-class-taut:4:d": ‹(φ ≡ ψ) → ((χ → φ) ≡ (χ → ψ))›
320 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
321 "→I" "→E" "&E" "&I" by metis
322AOT_theorem "oth-class-taut:4:e": ‹(φ ≡ ψ) → ((φ & χ) ≡ (ψ & χ))›
323 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
324 "→I" "→E" "&E" "&I" by metis
325AOT_theorem "oth-class-taut:4:f": ‹(φ ≡ ψ) → ((χ & φ) ≡ (χ & ψ))›
326 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
327 "→I" "→E" "&E" "&I" by metis
328
329AOT_theorem "oth-class-taut:4:g": ‹(φ ≡ ψ) ≡ ((φ & ψ) ∨ (¬φ & ¬ψ))›
330 apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
331 apply (drule "conventions:3"[THEN "df-rules-formulas[3]", THEN "→E"])
332 apply (metis "&I" "&E" "∨I"(1,2) MT(1) "raa-cor:3")
333 apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
334 using "&E" "∨E"(2) "raa-cor:3" by blast+
335AOT_theorem "oth-class-taut:4:h": ‹¬(φ ≡ ψ) ≡ ((φ & ¬ψ) ∨ (¬φ & ψ))›
336 apply (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I"; rule "→I")
337 apply (metis "&I" "∨I"(1, 2) "→I" MT(1) "df-rules-formulas[4]" "raa-cor:3" "conventions:3")
338 by (metis "&E" "∨E"(2) "→E" "df-rules-formulas[3]" "raa-cor:3" "conventions:3")
339AOT_theorem "oth-class-taut:5:a": ‹(φ & ψ) ≡ ¬(¬φ ∨ ¬ψ)›
340 using "conventions:3"[THEN "df-rules-formulas[4]"]
341 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
342AOT_theorem "oth-class-taut:5:b": ‹(φ ∨ ψ) ≡ ¬(¬φ & ¬ψ)›
343 using "conventions:3"[THEN "df-rules-formulas[4]"]
344 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
345AOT_theorem "oth-class-taut:5:c": ‹¬(φ & ψ) ≡ (¬φ ∨ ¬ψ)›
346 using "conventions:3"[THEN "df-rules-formulas[4]"]
347 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
348AOT_theorem "oth-class-taut:5:d": ‹¬(φ ∨ ψ) ≡ (¬φ & ¬ψ)›
349 using "conventions:3"[THEN "df-rules-formulas[4]"]
350 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
351
352lemmas DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"
353
354AOT_theorem "oth-class-taut:6:a": ‹(φ & (ψ ∨ χ)) ≡ ((φ & ψ) ∨ (φ & χ))›
355 using "conventions:3"[THEN "df-rules-formulas[4]"]
356 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
357AOT_theorem "oth-class-taut:6:b": ‹(φ ∨ (ψ & χ)) ≡ ((φ ∨ ψ) & (φ ∨ χ))›
358 using "conventions:3"[THEN "df-rules-formulas[4]"]
359 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
360
361AOT_theorem "oth-class-taut:7:a": ‹((φ & ψ) → χ) → (φ → (ψ → χ))›
362 by (metis "&I" "→E" "→I")
363lemmas Exportation = "oth-class-taut:7:a"
364AOT_theorem "oth-class-taut:7:b": ‹(φ → (ψ →χ)) → ((φ & ψ) → χ)›
365 by (metis "&E" "→E" "→I")
366lemmas Importation = "oth-class-taut:7:b"
367
368AOT_theorem "oth-class-taut:8:a": ‹(φ → (ψ → χ)) ≡ (ψ → (φ → χ))›
369 using "conventions:3"[THEN "df-rules-formulas[4]"] "→I" "→E" "&E" "&I" by metis
370lemmas Permutation = "oth-class-taut:8:a"
371AOT_theorem "oth-class-taut:8:b": ‹(φ → ψ) → ((φ → χ) → (φ → (ψ & χ)))›
372 by (metis "&I" "→E" "→I")
373lemmas Composition = "oth-class-taut:8:b"
374AOT_theorem "oth-class-taut:8:c": ‹(φ → χ) → ((ψ → χ) → ((φ ∨ ψ) → χ))›
375 by (metis "∨E"(2) "→E" "→I" RAA(1))
376AOT_theorem "oth-class-taut:8:d": ‹((φ → ψ) & (χ → Θ)) → ((φ & χ) → (ψ & Θ))›
377 by (metis "&E" "&I" "→E" "→I")
378lemmas "Double Composition" = "oth-class-taut:8:d"
379AOT_theorem "oth-class-taut:8:e": ‹((φ & ψ) ≡ (φ & χ)) ≡ (φ → (ψ ≡ χ))›
380 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
381 "→I" "→E" "&E" "&I")
382AOT_theorem "oth-class-taut:8:f": ‹((φ & ψ) ≡ (χ & ψ)) ≡ (ψ → (φ ≡ χ))›
383 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
384 "→I" "→E" "&E" "&I")
385AOT_theorem "oth-class-taut:8:g": ‹(ψ ≡ χ) → ((φ ∨ ψ) ≡ (φ ∨ χ))›
386 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
387 "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
388AOT_theorem "oth-class-taut:8:h": ‹(ψ ≡ χ) → ((ψ ∨ φ) ≡ (χ ∨ φ))›
389 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
390 "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
391AOT_theorem "oth-class-taut:8:i": ‹(φ ≡ (ψ & χ)) → (ψ → (φ ≡ χ))›
392 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
393 "→I" "→E" "&E" "&I")
394
395AOT_theorem "intro-elim:1": assumes ‹φ ∨ ψ› and ‹φ ≡ χ› and ‹ψ ≡ Θ› shows ‹χ ∨ Θ›
396 by (metis assms "∨I"(1, 2) "∨E"(1) "conventions:3"[THEN "df-rules-formulas[3]"] "→I" "→E" "&E"(1))
397
398AOT_theorem "intro-elim:2": assumes ‹φ → ψ› and ‹ψ → φ› shows ‹φ ≡ ψ›
399 by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
400lemmas "≡I" = "intro-elim:2"
401
402AOT_theorem "intro-elim:3:a": assumes ‹φ ≡ ψ› and ‹φ› shows ‹ψ›
403 by (metis "∨I"(1) "→I" "∨E"(1) "intro-elim:1" assms)
404AOT_theorem "intro-elim:3:b": assumes ‹φ ≡ ψ› and ‹ψ› shows ‹φ›
405 using "intro-elim:3:a" "Commutativity of ≡" assms by blast
406AOT_theorem "intro-elim:3:c": assumes ‹φ ≡ ψ› and ‹¬φ› shows ‹¬ψ›
407 using "intro-elim:3:b" "raa-cor:3" assms by blast
408AOT_theorem "intro-elim:3:d": assumes ‹φ ≡ ψ› and ‹¬ψ› shows ‹¬φ›
409 using "intro-elim:3:a" "raa-cor:3" assms by blast
410AOT_theorem "intro-elim:3:e": assumes ‹φ ≡ ψ› and ‹ψ ≡ χ› shows ‹φ ≡ χ›
411 by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
412declare "intro-elim:3:e"[trans]
413AOT_theorem "intro-elim:3:f": assumes ‹φ ≡ ψ› and ‹φ ≡ χ› shows ‹χ ≡ ψ›
414 by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
415lemmas "≡E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c" "intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"
416
417declare "Commutativity of ≡"[THEN "≡E"(1), sym]
418
419AOT_theorem "rule-eq-df:1": assumes ‹φ ≡⇩d⇩f ψ› shows ‹φ ≡ ψ›
420 by (simp add: "≡I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
421lemmas "≡Df" = "rule-eq-df:1"
422AOT_theorem "rule-eq-df:2": assumes ‹φ ≡⇩d⇩f ψ› and ‹φ› shows ‹ψ›
423 using "≡Df" "≡E"(1) assms by blast
424lemmas "≡⇩d⇩fE" = "rule-eq-df:2"
425AOT_theorem "rule-eq-df:3": assumes ‹φ ≡⇩d⇩f ψ› and ‹ψ› shows ‹φ›
426 using "≡Df" "≡E"(2) assms by blast
427lemmas "≡⇩d⇩fI" = "rule-eq-df:3"
428
429AOT_theorem "df-simplify:1": assumes ‹φ ≡ (ψ & χ)› and ‹ψ› shows ‹φ ≡ χ›
430 by (metis "&E"(2) "&I" "≡E"(1, 2) "≡I" "→I" assms)
431
432AOT_theorem "df-simplify:2": assumes ‹φ ≡ (ψ & χ)› and ‹χ› shows ‹φ ≡ ψ›
433 by (metis "&E"(1) "&I" "≡E"(1, 2) "≡I" "→I" assms)
434lemmas "≡S" = "df-simplify:1" "df-simplify:2"
435
436AOT_theorem "rule-ui:1": assumes ‹∀α φ{α}› and ‹τ↓› shows ‹φ{τ}›
437 using "→E" "cqt:1"[axiom_inst] assms by blast
438AOT_theorem "rule-ui:2[const_var]": assumes ‹∀α φ{α}› shows ‹φ{β}›
439 by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
440
441AOT_theorem "rule-ui:2[lambda]":
442 assumes ‹∀F φ{F}› and ‹INSTANCE_OF_CQT_2(ψ)›
443 shows ‹φ{[λν⇩1...ν⇩n ψ{ν⇩1...ν⇩n}]}›
444 by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
445AOT_theorem "rule-ui:3": assumes ‹∀α φ{α}› shows ‹φ{α}›
446 by (simp add: "rule-ui:2[const_var]" assms)
447lemmas "∀E" = "rule-ui:1" "rule-ui:2[const_var]" "rule-ui:2[lambda]" "rule-ui:3"
448
449AOT_theorem "cqt-orig:1[const_var]": ‹∀α φ{α} → φ{β}› by (simp add: "∀E"(2) "→I")
450AOT_theorem "cqt-orig:1[lambda]":
451 assumes ‹INSTANCE_OF_CQT_2(ψ)›
452 shows ‹∀F φ{F} → φ{[λν⇩1...ν⇩n ψ{ν⇩1...ν⇩n}]}›
453 by (simp add: "∀E"(3) "→I" assms)
454AOT_theorem "cqt-orig:2": ‹∀α (φ → ψ{α}) → (φ → ∀α ψ{α})›
455 by (metis "→I" GEN "vdash-properties:6" "∀E"(4))
456AOT_theorem "cqt-orig:3": ‹∀α φ{α} → φ{α}› using "cqt-orig:1[const_var]" .
457
458
459AOT_theorem universal: assumes ‹for arbitrary β: φ{β}› shows ‹∀α φ{α}›
460 using GEN assms .
461lemmas "∀I" = universal
462
463
464ML‹
465fun get_instantiated_allI ctxt varname thm = let
466val trm = Thm.concl_of thm
467val trm = case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
468 | _ => raise Term.TERM ("Expected simple theorem.", [trm])
469fun extractVars (Const (\<^const_name>‹AOT_term_of_var›, _) $ Var v) =
470 (if fst (fst v) = fst varname then [Var v] else [])
471 | extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
472 | extractVars (Abs (_, _, t)) = extractVars t
473 | extractVars _ = []
474val vars = extractVars trm
475val vars = fold Term.add_vars vars []
476val var = hd vars
477val trmty = case (snd var) of (Type (\<^type_name>‹AOT_var›, [t])) => (t)
478 | _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
479val trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
480 Const (\<^const_name>‹AOT_term_of_var›, Type ("fun", [snd var, trmty]))
481 $ Var var, trm))
482val trm = Thm.cterm_of (Context.proof_of ctxt) trm
483val ty = hd (Term.add_tvars (Thm.prop_of @{thm "∀I"}) [])
484val typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
485val allthm = Drule.instantiate_normalize ([(ty, typ)],[]) @{thm "∀I"}
486val phi = hd (Term.add_vars (Thm.prop_of allthm) [])
487val allthm = Drule.instantiate_normalize ([],[(phi,trm)]) allthm
488in
489allthm
490end
491›
492
493attribute_setup "∀I" =
494 ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
495 (fn ctxt => fn thm => fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm))›
496 "Quantify over a variable in a theorem using GEN."
497
498attribute_setup "unvarify" =
499 ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
500 (fn ctxt => fn thm =>
501 let
502 val thm = fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm
503 val thm = fold (fn _ => fn thm => thm RS @{thm "∀E"(1)}) args thm
504 in
505 thm
506 end))›
507 "Generalize a statement about variables to a statement about denoting terms."
508
509
510
511AOT_theorem "cqt-basic:1": ‹∀α∀β φ{α,β} ≡ ∀β∀α φ{α,β}›
512 by (metis "≡I" "∀E"(2) "∀I" "→I")
513
514AOT_theorem "cqt-basic:2": ‹∀α(φ{α} ≡ ψ{α}) ≡ (∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α}))›
515proof (rule "≡I"; rule "→I")
516 AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
517 AOT_hence ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
518 AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
519 using "≡E"(1,2) "→I" by blast+
520 AOT_thus ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
521 by (auto intro: "&I" "∀I")
522next
523 AOT_assume ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
524 AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
525 using "∀E"(2) "&E" by blast+
526 AOT_hence ‹φ{α} ≡ ψ{α}› for α
527 using "≡I" by blast
528 AOT_thus ‹∀α(φ{α} ≡ ψ{α})› by (auto intro: "∀I")
529qed
530
531AOT_theorem "cqt-basic:3": ‹∀α(φ{α} ≡ ψ{α}) → (∀α φ{α} ≡ ∀α ψ{α})›
532proof(rule "→I")
533 AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
534 AOT_hence 1: ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
535 {
536 AOT_assume ‹∀α φ{α}›
537 AOT_hence ‹∀α ψ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
538 }
539 moreover {
540 AOT_assume ‹∀α ψ{α}›
541 AOT_hence ‹∀α φ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
542 }
543 ultimately AOT_show ‹∀α φ{α} ≡ ∀α ψ{α}›
544 using "≡I" "→I" by auto
545qed
546
547AOT_theorem "cqt-basic:4": ‹∀α(φ{α} & ψ{α}) → (∀α φ{α} & ∀α ψ{α})›
548proof(rule "→I")
549 AOT_assume 0: ‹∀α(φ{α} & ψ{α})›
550 AOT_have ‹φ{α}› and ‹ψ{α}› for α using "∀E"(2) 0 "&E" by blast+
551 AOT_thus ‹∀α φ{α} & ∀α ψ{α}›
552 by (auto intro: "∀I" "&I")
553qed
554
555AOT_theorem "cqt-basic:5": ‹(∀α⇩1...∀α⇩n(φ{α⇩1...α⇩n})) → φ{α⇩1...α⇩n}›
556 using "cqt-orig:3" by blast
557
558AOT_theorem "cqt-basic:6": ‹∀α∀α φ{α} ≡ ∀α φ{α}›
559 by (meson "≡I" "→I" GEN "cqt-orig:1[const_var]")
560
561AOT_theorem "cqt-basic:7": ‹(φ → ∀α ψ{α}) ≡ ∀α(φ → ψ{α})›
562 by (metis "→I" "vdash-properties:6" "rule-ui:3" "≡I" GEN)
563
564AOT_theorem "cqt-basic:8": ‹(∀α φ{α} ∨ ∀α ψ{α}) → ∀α (φ{α} ∨ ψ{α})›
565 by (simp add: "∨I"(3) "→I" GEN "cqt-orig:1[const_var]")
566
567AOT_theorem "cqt-basic:9": ‹(∀α (φ{α} → ψ{α}) & ∀α (ψ{α} → χ{α})) → ∀α(φ{α} → χ{α})›
568proof -
569 {
570 AOT_assume ‹∀α (φ{α} → ψ{α})›
571 moreover AOT_assume ‹∀α (ψ{α} → χ{α})›
572 ultimately AOT_have ‹φ{α} → ψ{α}› and ‹ψ{α} → χ{α}› for α using "∀E" by blast+
573 AOT_hence ‹φ{α} → χ{α}› for α by (metis "→E" "→I")
574 AOT_hence ‹∀α(φ{α} → χ{α})› using "∀I" by fast
575 }
576 thus ?thesis using "&I" "→I" "&E" by meson
577qed
578
579AOT_theorem "cqt-basic:10": ‹(∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})) → ∀α (φ{α} ≡ χ{α})›
580proof(rule "→I"; rule "∀I")
581 fix β
582 AOT_assume ‹∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})›
583 AOT_hence ‹φ{β} ≡ ψ{β}› and ‹ψ{β} ≡ χ{β}› using "&E" "∀E" by blast+
584 AOT_thus ‹φ{β} ≡ χ{β}› using "≡I" "≡E" by blast
585qed
586
587AOT_theorem "cqt-basic:11": ‹∀α(φ{α} ≡ ψ{α}) ≡ ∀α (ψ{α} ≡ φ{α})›
588proof (rule "≡I"; rule "→I")
589 AOT_assume 0: ‹∀α(φ{α} ≡ ψ{α})›
590 {
591 fix α
592 AOT_have ‹φ{α} ≡ ψ{α}› using 0 "∀E" by blast
593 AOT_hence ‹ψ{α} ≡ φ{α}› using "≡I" "≡E" "→I" "→E" by metis
594 }
595 AOT_thus ‹∀α(ψ{α} ≡ φ{α})› using "∀I" by fast
596next
597 AOT_assume 0: ‹∀α(ψ{α} ≡ φ{α})›
598 {
599 fix α
600 AOT_have ‹ψ{α} ≡ φ{α}› using 0 "∀E" by blast
601 AOT_hence ‹φ{α} ≡ ψ{α}› using "≡I" "≡E" "→I" "→E" by metis
602 }
603 AOT_thus ‹∀α(φ{α} ≡ ψ{α})› using "∀I" by fast
604qed
605
606AOT_theorem "cqt-basic:12": ‹∀α φ{α} → ∀α (ψ{α} → φ{α})›
607 by (simp add: "∀E"(2) "→I" GEN)
608
609AOT_theorem "cqt-basic:13": ‹∀α φ{α} ≡ ∀β φ{β}›
610 using "≡I" "→I" by blast
611
612AOT_theorem "cqt-basic:14": ‹(∀α⇩1...∀α⇩n (φ{α⇩1...α⇩n} → ψ{α⇩1...α⇩n})) → ((∀α⇩1...∀α⇩n φ{α⇩1...α⇩n}) → (∀α⇩1...∀α⇩n ψ{α⇩1...α⇩n}))›
613 using "cqt:3"[axiom_inst] by auto
614
615AOT_theorem "cqt-basic:15": ‹(∀α⇩1...∀α⇩n (φ → ψ{α⇩1...α⇩n})) → (φ → (∀α⇩1...∀α⇩n ψ{α⇩1...α⇩n}))›
616 using "cqt-orig:2" by auto
617
618
619AOT_theorem "universal-cor": assumes ‹for arbitrary β: φ{β}› shows ‹∀α φ{α}›
620 using GEN assms .
621
622AOT_theorem "existential:1": assumes ‹φ{τ}› and ‹τ↓› shows ‹∃α φ{α}›
623proof(rule "raa-cor:1")
624 AOT_assume ‹¬∃α φ{α}›
625 AOT_hence ‹∀α ¬φ{α}›
626 using "≡⇩d⇩fI" "conventions:4" RAA "&I" by blast
627 AOT_hence ‹¬φ{τ}› using assms(2) "∀E"(1) "→E" by blast
628 AOT_thus ‹φ{τ} & ¬φ{τ}› using assms(1) "&I" by blast
629qed
630
631AOT_theorem "existential:2[const_var]": assumes ‹φ{β}› shows ‹∃α φ{α}›
632 using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast
633
634AOT_theorem "existential:2[lambda]":
635 assumes ‹φ{[λν⇩1...ν⇩n ψ{ν⇩1...ν⇩n}]}› and ‹INSTANCE_OF_CQT_2(ψ)›
636 shows ‹∃α φ{α}›
637 using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
638lemmas "∃I" = "existential:1" "existential:2[const_var]" "existential:2[lambda]"
639
640AOT_theorem "instantiation":
641 assumes ‹for arbitrary β: φ{β} ❙⊢ ψ› and ‹∃α φ{α}›
642 shows ‹ψ›
643 by (metis (no_types, lifting) "≡⇩d⇩fE" GEN "raa-cor:3" "conventions:4" assms)
644lemmas "∃E" = "instantiation"
645
646AOT_theorem "cqt-further:1": ‹∀α φ{α} → ∃α φ{α}›
647 using "∀E"(4) "∃I"(2) "→I" by metis
648
649AOT_theorem "cqt-further:2": ‹¬∀α φ{α} → ∃α ¬φ{α}›
650 using "∀I" "∃I"(2) "→I" RAA by metis
651
652AOT_theorem "cqt-further:3": ‹∀α φ{α} ≡ ¬∃α ¬φ{α}›
653 using "∀E"(4) "∃E" "→I" RAA
654 by (metis "cqt-further:2" "≡I" "modus-tollens:1")
655
656AOT_theorem "cqt-further:4": ‹¬∃α φ{α} → ∀α ¬φ{α}›
657 using "∀I" "∃I"(2)"→I" RAA by metis
658
659AOT_theorem "cqt-further:5": ‹∃α (φ{α} & ψ{α}) → (∃α φ{α} & ∃α ψ{α})›
660 by (metis (no_types, lifting) "&E" "&I" "∃E" "∃I"(2) "→I")
661
662AOT_theorem "cqt-further:6": ‹∃α (φ{α} ∨ ψ{α}) → (∃α φ{α} ∨ ∃α ψ{α})›
663 by (metis (mono_tags, lifting) "∃E" "∃I"(2) "∨E"(3) "∨I"(1, 2) "→I" RAA(2))
664
665AOT_theorem "cqt-further:7": ‹∃α φ{α} ≡ ∃β φ{β}›
666 by (simp add: "oth-class-taut:3:a")
667
668AOT_theorem "cqt-further:8": ‹(∀α φ{α} & ∀α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
669 by (metis (mono_tags, lifting) "&E" "≡I" "∀E"(2) "→I" GEN)
670
671AOT_theorem "cqt-further:9": ‹(¬∃α φ{α} & ¬∃α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
672 by (metis (mono_tags, lifting) "&E" "≡I" "∃I"(2) "→I" GEN "raa-cor:4")
673
674AOT_theorem "cqt-further:10": ‹(∃α φ{α} & ¬∃α ψ{α}) → ¬∀α (φ{α} ≡ ψ{α})›
675proof(rule "→I"; rule "raa-cor:2")
676 AOT_assume 0: ‹∃α φ{α} & ¬∃α ψ{α}›
677 then AOT_obtain α where ‹φ{α}› using "∃E" "&E"(1) by metis
678 moreover AOT_assume ‹∀α (φ{α} ≡ ψ{α})›
679 ultimately AOT_have ‹ψ{α}› using "∀E"(4) "≡E"(1) by blast
680 AOT_hence ‹∃α ψ{α}› using "∃I" by blast
681 AOT_thus ‹∃α ψ{α} & ¬∃α ψ{α}› using 0 "&E"(2) "&I" by blast
682qed
683
684AOT_theorem "cqt-further:11": ‹∃α∃β φ{α,β} ≡ ∃β∃α φ{α,β}›
685 using "≡I" "→I" "∃I"(2) "∃E" by metis
686
687AOT_theorem "log-prop-prop:1": ‹[λ φ]↓›
688 using "cqt:2[lambda0]"[axiom_inst] by auto
689
690AOT_theorem "log-prop-prop:2": ‹φ↓›
691 by (rule "≡⇩d⇩fI"[OF "existence:3"]) "cqt:2[lambda]"
692
693AOT_theorem "exist-nec": ‹τ↓ → □τ↓›
694proof -
695 AOT_have ‹∀β □β↓›
696 by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
697 AOT_thus ‹τ↓ → □τ↓›
698 using "cqt:1"[axiom_inst] "→E" by blast
699qed
700
701
702class AOT_Term_id = AOT_Term +
703 assumes "t=t-proper:1"[AOT]: ‹[v ⊨ τ = τ' → τ↓]›
704 and "t=t-proper:2"[AOT]: ‹[v ⊨ τ = τ' → τ'↓]›
705
706instance κ :: AOT_Term_id
707proof
708 AOT_modally_strict {
709 AOT_show ‹κ = κ' → κ↓› for κ κ'
710 proof(rule "→I")
711 AOT_assume ‹κ = κ'›
712 AOT_hence ‹O!κ ∨ A!κ›
713 by (rule "∨I"(3)[OF "≡⇩d⇩fE"[OF "identity:1"]])
714 (meson "→I" "∨I"(1) "&E"(1))+
715 AOT_thus ‹κ↓›
716 by (rule "∨E"(1))
717 (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
718 qed
719 }
720next
721 AOT_modally_strict {
722 AOT_show ‹κ = κ' → κ'↓› for κ κ'
723 proof(rule "→I")
724 AOT_assume ‹κ = κ'›
725 AOT_hence ‹O!κ' ∨ A!κ'›
726 by (rule "∨I"(3)[OF "≡⇩d⇩fE"[OF "identity:1"]])
727 (meson "→I" "∨I" "&E")+
728 AOT_thus ‹κ'↓›
729 by (rule "∨E"(1))
730 (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
731 qed
732 }
733qed
734
735instance rel :: (AOT_κs) AOT_Term_id
736proof
737 AOT_modally_strict {
738 AOT_show ‹Π = Π' → Π↓› for Π Π' :: ‹<'a>›
739 proof(rule "→I")
740 AOT_assume ‹Π = Π'›
741 AOT_thus ‹Π↓› using "≡⇩d⇩fE"[OF "identity:3"[of Π Π']] "&E" by blast
742 qed
743 }
744next
745 AOT_modally_strict {
746 AOT_show ‹Π = Π' → Π'↓› for Π Π' :: ‹<'a>›
747 proof(rule "→I")
748 AOT_assume ‹Π = Π'›
749 AOT_thus ‹Π'↓› using "≡⇩d⇩fE"[OF "identity:3"[of Π Π']] "&E" by blast
750 qed
751 }
752qed
753
754instance 𝗈 :: AOT_Term_id
755proof
756 AOT_modally_strict {
757 fix φ ψ
758 AOT_show ‹φ = ψ → φ↓›
759 proof(rule "→I")
760 AOT_assume ‹φ = ψ›
761 AOT_thus ‹φ↓› using "≡⇩d⇩fE"[OF "identity:4"[of φ ψ]] "&E" by blast
762 qed
763 }
764next
765 AOT_modally_strict {
766 fix φ ψ
767 AOT_show ‹φ = ψ → ψ↓›
768 proof(rule "→I")
769 AOT_assume ‹φ = ψ›
770 AOT_thus ‹ψ↓› using "≡⇩d⇩fE"[OF "identity:4"[of φ ψ]] "&E" by blast
771 qed
772 }
773qed
774
775instance prod :: (AOT_Term_id, AOT_Term_id) AOT_Term_id
776proof
777 AOT_modally_strict {
778 fix τ τ' :: ‹'a×'b›
779 AOT_show ‹τ = τ' → τ↓›
780 proof (induct τ; induct τ'; rule "→I")
781 fix τ⇩1 τ⇩1' :: 'a and τ⇩2 τ⇩2' :: 'b
782 AOT_assume ‹«(τ⇩1, τ⇩2)» = «(τ⇩1', τ⇩2')»›
783 AOT_hence ‹(τ⇩1 = τ⇩1') & (τ⇩2 = τ⇩2')› by (metis "≡⇩d⇩fE" tuple_identity_1)
784 AOT_hence ‹τ⇩1↓› and ‹τ⇩2↓› using "t=t-proper:1" "&E" "vdash-properties:10" by blast+
785 AOT_thus ‹«(τ⇩1, τ⇩2)»↓› by (metis "≡⇩d⇩fI" "&I" tuple_denotes)
786 qed
787 }
788next
789 AOT_modally_strict {
790 fix τ τ' :: ‹'a×'b›
791 AOT_show ‹τ = τ' → τ'↓›
792 proof (induct τ; induct τ'; rule "→I")
793 fix τ⇩1 τ⇩1' :: 'a and τ⇩2 τ⇩2' :: 'b
794 AOT_assume ‹«(τ⇩1, τ⇩2)» = «(τ⇩1', τ⇩2')»›
795 AOT_hence ‹(τ⇩1 = τ⇩1') & (τ⇩2 = τ⇩2')› by (metis "≡⇩d⇩fE" tuple_identity_1)
796 AOT_hence ‹τ⇩1'↓› and ‹τ⇩2'↓› using "t=t-proper:2" "&E" "vdash-properties:10" by blast+
797 AOT_thus ‹«(τ⇩1', τ⇩2')»↓› by (metis "≡⇩d⇩fI" "&I" tuple_denotes)
798 qed
799 }
800qed
801
802
803AOT_register_type_constraints
804 Term: ‹_::AOT_Term_id› ‹_::AOT_Term_id›
805AOT_register_type_constraints
806 Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id}›
807AOT_register_type_constraints
808 Relation: ‹<_::{AOT_κs, AOT_Term_id}>›
809
810AOT_theorem "id-rel-nec-equiv:1": ‹Π = Π' → □∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π']x⇩1...x⇩n)›
811proof(rule "→I")
812 AOT_assume assumption: ‹Π = Π'›
813 AOT_hence ‹Π↓› and ‹Π'↓›
814 using "t=t-proper:1" "t=t-proper:2" MP by blast+
815 moreover AOT_have ‹∀F∀G (F = G → ((□∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [F]x⇩1...x⇩n)) → □∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)))›
816 apply (rule GEN)+ using "l-identity"[axiom_inst] by force
817 ultimately AOT_have ‹Π = Π' → ((□∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π]x⇩1...x⇩n)) → □∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π']x⇩1...x⇩n))›
818 using "∀E"(1) by blast
819 AOT_hence ‹(□∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π]x⇩1...x⇩n)) → □∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π']x⇩1...x⇩n)›
820 using assumption "→E" by blast
821 moreover AOT_have ‹□∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π]x⇩1...x⇩n)›
822 by (simp add: RN "oth-class-taut:3:a" "universal-cor")
823 ultimately AOT_show ‹□∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π']x⇩1...x⇩n)›
824 using "→E" by blast
825qed
826
827AOT_theorem "id-rel-nec-equiv:2": ‹φ = ψ → □(φ ≡ ψ)›
828proof(rule "→I")
829 AOT_assume assumption: ‹φ = ψ›
830 AOT_hence ‹φ↓› and ‹ψ↓›
831 using "t=t-proper:1" "t=t-proper:2" MP by blast+
832 moreover AOT_have ‹∀p∀q (p = q → ((□(p ≡ p) → □(p ≡ q))))›
833 apply (rule GEN)+ using "l-identity"[axiom_inst] by force
834 ultimately AOT_have ‹φ = ψ → (□(φ ≡ φ) → □(φ ≡ ψ))›
835 using "∀E"(1) by blast
836 AOT_hence ‹□(φ ≡ φ) → □(φ ≡ ψ)›
837 using assumption "→E" by blast
838 moreover AOT_have ‹□(φ ≡ φ)›
839 by (simp add: RN "oth-class-taut:3:a" "universal-cor")
840 ultimately AOT_show ‹□(φ ≡ ψ)›
841 using "→E" by blast
842qed
843
844AOT_theorem "rule=E": assumes ‹φ{τ}› and ‹τ = σ› shows ‹φ{σ}›
845proof -
846 AOT_have ‹τ↓› and ‹σ↓› using assms(2) "t=t-proper:1" "t=t-proper:2" "→E" by blast+
847 moreover AOT_have ‹∀α∀β(α = β → (φ{α} → φ{β}))›
848 apply (rule GEN)+ using "l-identity"[axiom_inst] by blast
849 ultimately AOT_have ‹τ = σ → (φ{τ} → φ{σ})›
850 using "∀E"(1) by blast
851 AOT_thus ‹φ{σ}› using assms "→E" by blast
852qed
853
854AOT_theorem "propositions-lemma:1": ‹[λ φ] = φ›
855proof -
856 AOT_have ‹φ↓› by (simp add: "log-prop-prop:2")
857 moreover AOT_have ‹∀p [λ p] = p› using "lambda-predicates:3[zero]"[axiom_inst] "∀I" by fast
858 ultimately AOT_show ‹[λ φ] = φ›
859 using "∀E" by blast
860qed
861
862AOT_theorem "propositions-lemma:2": ‹[λ φ] ≡ φ›
863proof -
864 AOT_have ‹[λ φ] ≡ [λ φ]› by (simp add: "oth-class-taut:3:a")
865 AOT_thus ‹[λ φ] ≡ φ› using "propositions-lemma:1" "rule=E" by blast
866qed
867
868
869
870AOT_theorem "propositions-lemma:6": ‹(φ ≡ ψ) ≡ ([λ φ] ≡ [λ ψ])›
871 by (metis "≡E"(1) "≡E"(5) "Associativity of ≡" "propositions-lemma:2")
872
873
874
875AOT_theorem "oa-exist:1": ‹O!↓›
876proof -
877 AOT_have ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
878 AOT_hence 1: ‹O! = [λx ◇[E!]x]› using "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1)] "→E" by blast
879 AOT_show ‹O!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
880qed
881
882AOT_theorem "oa-exist:2": ‹A!↓›
883proof -
884 AOT_have ‹[λx ¬◇[E!]x]↓› by "cqt:2[lambda]"
885 AOT_hence 1: ‹A! = [λx ¬◇[E!]x]› using "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1)] "→E" by blast
886 AOT_show ‹A!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
887qed
888
889AOT_theorem "oa-exist:3": ‹O!x ∨ A!x›
890proof(rule "raa-cor:1")
891 AOT_assume ‹¬(O!x ∨ A!x)›
892 AOT_hence A: ‹¬O!x› and B: ‹¬A!x›
893 using "Disjunction Addition"(1) "modus-tollens:1" "∨I"(2) "raa-cor:5" by blast+
894 AOT_have C: ‹O! = [λx ◇[E!]x]›
895 by (rule "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
896 AOT_have D: ‹A! = [λx ¬◇[E!]x]›
897 by (rule "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
898 AOT_have E: ‹¬[λx ◇[E!]x]x›
899 using A C "rule=E" by fast
900 AOT_have F: ‹¬[λx ¬◇[E!]x]x›
901 using B D "rule=E" by fast
902 AOT_have G: ‹[λx ◇[E!]x]x ≡ ◇[E!]x›
903 by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
904 AOT_have H: ‹[λx ¬◇[E!]x]x ≡ ¬◇[E!]x›
905 by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
906 AOT_show ‹¬◇[E!]x & ¬¬◇[E!]x› using G E "≡E" H F "≡E" "&I" by metis
907qed
908
909AOT_theorem "p-identity-thm2:1": ‹F = G ≡ □∀x(x[F] ≡ x[G])›
910proof -
911 AOT_have ‹F = G ≡ F↓ & G↓ & □∀x(x[F] ≡ x[G])›
912 using "identity:2" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
913 moreover AOT_have ‹F↓› and ‹G↓›
914 by (auto simp: "cqt:2[const_var]"[axiom_inst])
915 ultimately AOT_show ‹F = G ≡ □∀x(x[F] ≡ x[G])›
916 using "≡S"(1) "&I" by blast
917qed
918
919AOT_theorem "p-identity-thm2:2[2]": ‹F = G ≡ ∀y⇩1([λx [F]xy⇩1] = [λx [G]xy⇩1] & [λx [F]y⇩1x] = [λx [G]y⇩1x])›
920proof -
921 AOT_have ‹F = G ≡ F↓ & G↓ & ∀y⇩1([λx [F]xy⇩1] = [λx [G]xy⇩1] & [λx [F]y⇩1x] = [λx [G]y⇩1x])›
922 using "identity:3[2]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
923 moreover AOT_have ‹F↓› and ‹G↓›
924 by (auto simp: "cqt:2[const_var]"[axiom_inst])
925 ultimately show ?thesis
926 using "≡S"(1) "&I" by blast
927qed
928
929AOT_theorem "p-identity-thm2:2[3]": ‹F = G ≡ ∀y⇩1∀y⇩2([λx [F]xy⇩1y⇩2] = [λx [G]xy⇩1y⇩2] & [λx [F]y⇩1xy⇩2] = [λx [G]y⇩1xy⇩2] & [λx [F]y⇩1y⇩2x] = [λx [G]y⇩1y⇩2x])›
930proof -
931 AOT_have ‹F = G ≡ F↓ & G↓ & ∀y⇩1∀y⇩2([λx [F]xy⇩1y⇩2] = [λx [G]xy⇩1y⇩2] & [λx [F]y⇩1xy⇩2] = [λx [G]y⇩1xy⇩2] & [λx [F]y⇩1y⇩2x] = [λx [G]y⇩1y⇩2x])›
932 using "identity:3[3]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
933 moreover AOT_have ‹F↓› and ‹G↓›
934 by (auto simp: "cqt:2[const_var]"[axiom_inst])
935 ultimately show ?thesis
936 using "≡S"(1) "&I" by blast
937qed
938
939AOT_theorem "p-identity-thm2:2[4]": ‹F = G ≡ ∀y⇩1∀y⇩2∀y⇩3([λx [F]xy⇩1y⇩2y⇩3] = [λx [G]xy⇩1y⇩2y⇩3] & [λx [F]y⇩1xy⇩2y⇩3] = [λx [G]y⇩1xy⇩2y⇩3] & [λx [F]y⇩1y⇩2xy⇩3] = [λx [G]y⇩1y⇩2xy⇩3] & [λx [F]y⇩1y⇩2y⇩3x] = [λx [G]y⇩1y⇩2y⇩3x])›
940proof -
941 AOT_have ‹F = G ≡ F↓ & G↓ & ∀y⇩1∀y⇩2∀y⇩3([λx [F]xy⇩1y⇩2y⇩3] = [λx [G]xy⇩1y⇩2y⇩3] & [λx [F]y⇩1xy⇩2y⇩3] = [λx [G]y⇩1xy⇩2y⇩3] & [λx [F]y⇩1y⇩2xy⇩3] = [λx [G]y⇩1y⇩2xy⇩3] & [λx [F]y⇩1y⇩2y⇩3x] = [λx [G]y⇩1y⇩2y⇩3x])›
942 using "identity:3[4]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
943 moreover AOT_have ‹F↓› and ‹G↓›
944 by (auto simp: "cqt:2[const_var]"[axiom_inst])
945 ultimately show ?thesis
946 using "≡S"(1) "&I" by blast
947qed
948
949AOT_theorem "p-identity-thm2:2":
950 ‹F = G ≡ ∀x⇩1...∀x⇩n «AOT_sem_proj_id x⇩1x⇩n (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
951proof -
952 AOT_have ‹F = G ≡ F↓ & G↓ & ∀x⇩1...∀x⇩n «AOT_sem_proj_id x⇩1x⇩n (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
953 using "identity:3" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
954 moreover AOT_have ‹F↓› and ‹G↓›
955 by (auto simp: "cqt:2[const_var]"[axiom_inst])
956 ultimately show ?thesis
957 using "≡S"(1) "&I" by blast
958qed
959
960AOT_theorem "p-identity-thm2:3":
961 ‹p = q ≡ [λx p] = [λx q]›
962proof -
963 AOT_have ‹p = q ≡ p↓ & q↓ & [λx p] = [λx q]›
964 using "identity:4" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
965 moreover AOT_have ‹p↓› and ‹q↓›
966 by (auto simp: "cqt:2[const_var]"[axiom_inst])
967 ultimately show ?thesis
968 using "≡S"(1) "&I" by blast
969qed
970
971class AOT_Term_id_2 = AOT_Term_id + assumes "id-eq:1": ‹[v ⊨ α = α]›
972
973instance κ :: AOT_Term_id_2
974proof
975 AOT_modally_strict {
976 fix x
977 {
978 AOT_assume ‹O!x›
979 moreover AOT_have ‹□∀F([F]x ≡ [F]x)›
980 using RN GEN "oth-class-taut:3:a" by fast
981 ultimately AOT_have ‹O!x & O!x & □∀F([F]x ≡ [F]x)› using "&I" by simp
982 }
983 moreover {
984 AOT_assume ‹A!x›
985 moreover AOT_have ‹□∀F(x[F] ≡ x[F])›
986 using RN GEN "oth-class-taut:3:a" by fast
987 ultimately AOT_have ‹A!x & A!x & □∀F(x[F] ≡ x[F])› using "&I" by simp
988 }
989 ultimately AOT_have ‹(O!x & O!x & □∀F([F]x ≡ [F]x)) ∨ (A!x & A!x & □∀F(x[F] ≡ x[F]))›
990 using "oa-exist:3" "∨I"(1) "∨I"(2) "∨E"(3) "raa-cor:1" by blast
991 AOT_thus ‹x = x›
992 using "identity:1"[THEN "df-rules-formulas[4]"] "→E" by blast
993 }
994qed
995
996instance rel :: ("{AOT_κs,AOT_Term_id_2}") AOT_Term_id_2
997proof
998 AOT_modally_strict {
999 fix F :: "<'a> AOT_var"
1000 AOT_have 0: ‹[λx⇩1...x⇩n [F]x⇩1...x⇩n] = F›
1001 by (simp add: "lambda-predicates:3"[axiom_inst])
1002 AOT_have ‹[λx⇩1...x⇩n [F]x⇩1...x⇩n]↓›
1003 by "cqt:2[lambda]"
1004 AOT_hence ‹[λx⇩1...x⇩n [F]x⇩1...x⇩n] = [λx⇩1...x⇩n [F]x⇩1...x⇩n]›
1005 using "lambda-predicates:1"[axiom_inst] "→E" by blast
1006 AOT_show ‹F = F› using "rule=E" 0 by force
1007 }
1008qed
1009
1010instance 𝗈 :: AOT_Term_id_2
1011proof
1012 AOT_modally_strict {
1013 fix p
1014 AOT_have 0: ‹[λ p] = p›
1015 by (simp add: "lambda-predicates:3[zero]"[axiom_inst])
1016 AOT_have ‹[λ p]↓›
1017 by (rule "cqt:2[lambda0]"[axiom_inst])
1018 AOT_hence ‹[λ p] = [λ p]›
1019 using "lambda-predicates:1[zero]"[axiom_inst] "→E" by blast
1020 AOT_show ‹p = p› using "rule=E" 0 by force
1021 }
1022qed
1023
1024instance prod :: (AOT_Term_id_2, AOT_Term_id_2) AOT_Term_id_2
1025proof
1026 AOT_modally_strict {
1027 fix α :: ‹('a×'b) AOT_var›
1028 AOT_show ‹α = α›
1029 proof (induct)
1030 AOT_show ‹τ = τ› if ‹τ↓› for τ :: ‹'a×'b›
1031 using that
1032 proof (induct τ)
1033 fix τ⇩1 :: 'a and τ⇩2 :: 'b
1034 AOT_assume ‹«(τ⇩1,τ⇩2)»↓›
1035 AOT_hence ‹τ⇩1↓› and ‹τ⇩2↓› using "≡⇩d⇩fE" "&E" tuple_denotes by blast+
1036 AOT_hence ‹τ⇩1 = τ⇩1› and ‹τ⇩2 = τ⇩2› using "id-eq:1"[unvarify α] by blast+
1037 AOT_thus ‹«(τ⇩1, τ⇩2)» = «(τ⇩1, τ⇩2)»› by (metis "≡⇩d⇩fI" "&I" tuple_identity_1)
1038 qed
1039 qed
1040 }
1041qed
1042
1043AOT_register_type_constraints
1044 Term: ‹_::AOT_Term_id_2› ‹_::AOT_Term_id_2›
1045AOT_register_type_constraints
1046 Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id_2}›
1047AOT_register_type_constraints
1048 Relation: ‹<_::{AOT_κs, AOT_Term_id_2}>›
1049
1050
1051AOT_theorem "id-eq:2": ‹α = β → β = α›
1052
1062
1063proof (rule "→I")
1064 AOT_assume ‹α = β›
1065 moreover AOT_have ‹β = β› using calculation "rule=E"[of _ "λ τ . «τ = β»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1066 moreover AOT_have ‹α = α → α = α› using "if-p-then-p" by blast
1067 ultimately AOT_show ‹β = α›
1068 using "→E" "→I" "rule=E"[of _ "λ τ . «(τ = τ) → (τ = α)»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1069qed
1070
1071AOT_theorem "id-eq:3": ‹α = β & β = γ → α = γ›
1072 using "rule=E" "→I" "&E" by blast
1073
1074AOT_theorem "id-eq:4": ‹α = β ≡ ∀γ (α = γ ≡ β = γ)›
1075proof (rule "≡I"; rule "→I")
1076 AOT_assume 0: ‹α = β›
1077 AOT_hence 1: ‹β = α› using "id-eq:2" "→E" by blast
1078 AOT_show ‹∀γ (α = γ ≡ β = γ)›
1079 by (rule GEN) (metis "≡I" "→I" 0 "1" "rule=E")
1080next
1081 AOT_assume ‹∀γ (α = γ ≡ β = γ)›
1082 AOT_hence ‹α = α ≡ β = α› using "∀E"(2) by blast
1083 AOT_hence ‹α = α → β = α› using "≡E"(1) "→I" by blast
1084 AOT_hence ‹β = α› using "id-eq:1" "→E" by blast
1085 AOT_thus ‹α = β› using "id-eq:2" "→E" by blast
1086qed
1087
1088AOT_theorem "rule=I:1": assumes ‹τ↓› shows ‹τ = τ›
1089proof -
1090 AOT_have ‹∀α (α = α)›
1091 by (rule GEN) (metis "id-eq:1")
1092 AOT_thus ‹τ = τ› using assms "∀E" by blast
1093qed
1094
1095AOT_theorem "rule=I:2[const_var]": "α = α"
1096 using "id-eq:1".
1097
1098AOT_theorem "rule=I:2[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows "[λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}] = [λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}]"
1099proof -
1100 AOT_have ‹∀α (α = α)›
1101 by (rule GEN) (metis "id-eq:1")
1102 moreover AOT_have ‹[λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}]↓› using assms by (rule "cqt:2[lambda]"[axiom_inst])
1103 ultimately AOT_show ‹[λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}] = [λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}]› using assms "∀E" by blast
1104qed
1105
1106lemmas "=I" = "rule=I:1" "rule=I:2[const_var]" "rule=I:2[lambda]"
1107
1108AOT_theorem "rule-id-def:1":
1109 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}› and ‹σ{τ⇩1...τ⇩n}↓›
1110 shows ‹τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}›
1111proof -
1112 AOT_have ‹σ{τ⇩1...τ⇩n}↓ → τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}›
1113 using "df-rules-terms[3]" assms(1) "&E" by blast
1114 AOT_thus ‹τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}›
1115 using assms(2) "→E" by blast
1116qed
1117
1118AOT_theorem "rule-id-def:1[zero]":
1119 assumes ‹τ =⇩d⇩f σ› and ‹σ↓›
1120 shows ‹τ = σ›
1121proof -
1122 AOT_have ‹σ↓ → τ = σ›
1123 using "df-rules-terms[4]" assms(1) "&E" by blast
1124 AOT_thus ‹τ = σ›
1125 using assms(2) "→E" by blast
1126qed
1127
1128AOT_theorem "rule-id-def:2:a":
1129 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}› and ‹σ{τ⇩1...τ⇩n}↓› and ‹φ{τ{τ⇩1...τ⇩n}}›
1130 shows ‹φ{σ{τ⇩1...τ⇩n}}›
1131proof -
1132 AOT_have ‹τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}› using "rule-id-def:1" assms(1,2) by blast
1133 AOT_thus ‹φ{σ{τ⇩1...τ⇩n}}› using assms(3) "rule=E" by blast
1134qed
1135
1136
1137AOT_theorem "rule-id-def:2:a[2]":
1138 assumes ‹τ{«(α⇩1,α⇩2)»} =⇩d⇩f σ{«(α⇩1,α⇩2)»}› and ‹σ{«(τ⇩1,τ⇩2)»}↓› and ‹φ{τ{«(τ⇩1,τ⇩2)»}}›
1139 shows ‹φ{σ{«(τ⇩1,τ⇩2)»}}›
1140proof -
1141 AOT_have ‹τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1142 proof -
1143 AOT_have ‹σ{«(τ⇩1,τ⇩2)»}↓ → τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1144 using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def)
1145 AOT_thus ‹τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1146 using assms(2) "→E" by blast
1147 qed
1148 AOT_thus ‹φ{σ{«(τ⇩1,τ⇩2)»}}› using assms(3) "rule=E" by blast
1149qed
1150
1151AOT_theorem "rule-id-def:2:a[zero]":
1152 assumes ‹τ =⇩d⇩f σ› and ‹σ↓› and ‹φ{τ}›
1153 shows ‹φ{σ}›
1154proof -
1155 AOT_have ‹τ = σ› using "rule-id-def:1[zero]" assms(1,2) by blast
1156 AOT_thus ‹φ{σ}› using assms(3) "rule=E" by blast
1157qed
1158
1159lemmas "=⇩d⇩fE" = "rule-id-def:2:a" "rule-id-def:2:a[zero]"
1160
1161AOT_theorem "rule-id-def:2:b":
1162 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}› and ‹σ{τ⇩1...τ⇩n}↓› and ‹φ{σ{τ⇩1...τ⇩n}}›
1163 shows ‹φ{τ{τ⇩1...τ⇩n}}›
1164proof -
1165 AOT_have ‹τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}› using "rule-id-def:1" assms(1,2) by blast
1166 AOT_hence ‹σ{τ⇩1...τ⇩n} = τ{τ⇩1...τ⇩n}›
1167 using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1168 AOT_thus ‹φ{τ{τ⇩1...τ⇩n}}› using assms(3) "rule=E" by blast
1169qed
1170
1171
1172AOT_theorem "rule-id-def:2:b[2]":
1173 assumes ‹τ{«(α⇩1,α⇩2)»} =⇩d⇩f σ{«(α⇩1,α⇩2)»}› and ‹σ{«(τ⇩1,τ⇩2)»}↓› and ‹φ{σ{«(τ⇩1,τ⇩2)»}}›
1174 shows ‹φ{τ{«(τ⇩1,τ⇩2)»}}›
1175proof -
1176 AOT_have ‹τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1177 proof -
1178 AOT_have ‹σ{«(τ⇩1,τ⇩2)»}↓ → τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1179 using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def)
1180 AOT_thus ‹τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1181 using assms(2) "→E" by blast
1182 qed
1183 AOT_hence ‹σ{«(τ⇩1,τ⇩2)»} = τ{«(τ⇩1,τ⇩2)»}›
1184 using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1185 AOT_thus ‹φ{τ{«(τ⇩1,τ⇩2)»}}› using assms(3) "rule=E" by blast
1186qed
1187
1188AOT_theorem "rule-id-def:2:b[zero]":
1189 assumes ‹τ =⇩d⇩f σ› and ‹σ↓› and ‹φ{σ}›
1190 shows ‹φ{τ}›
1191proof -
1192 AOT_have ‹τ = σ› using "rule-id-def:1[zero]" assms(1,2) by blast
1193 AOT_hence ‹σ = τ›
1194 using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1195 AOT_thus ‹φ{τ}› using assms(3) "rule=E" by blast
1196qed
1197
1198lemmas "=⇩d⇩fI" = "rule-id-def:2:b" "rule-id-def:2:b[zero]"
1199
1200AOT_theorem "free-thms:1": ‹τ↓ ≡ ∃β (β = τ)›
1201 by (metis "∃E" "rule=I:1" "t=t-proper:2" "→I" "∃I"(1) "≡I" "→E")
1202
1203AOT_theorem "free-thms:2": ‹∀α φ{α} → (∃β (β = τ) → φ{τ})›
1204 by (metis "∃E" "rule=E" "cqt:2[const_var]"[axiom_inst] "→I" "∀E"(1))
1205
1206AOT_theorem "free-thms:3[const_var]": ‹∃β (β = α)›
1207 by (meson "∃I"(2) "id-eq:1")
1208
1209AOT_theorem "free-thms:3[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows ‹∃β (β = [λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}])›
1210 by (meson "=I"(3) assms "cqt:2[lambda]"[axiom_inst] "existential:1")
1211
1212AOT_theorem "free-thms:4[rel]": ‹([Π]κ⇩1...κ⇩n ∨ κ⇩1...κ⇩n[Π]) → ∃β (β = Π)›
1213 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1214
1215
1217AOT_theorem "free-thms:4[vars]": ‹([Π]κ⇩1...κ⇩n ∨ κ⇩1...κ⇩n[Π]) → ∃β⇩1...∃β⇩n (β⇩1...β⇩n = κ⇩1...κ⇩n)›
1218 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1219
1220AOT_theorem "free-thms:4[1,rel]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = Π)›
1221 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1222AOT_theorem "free-thms:4[1,1]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = κ)›
1223 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1224
1225AOT_theorem "free-thms:4[2,rel]": ‹([Π]κ⇩1κ⇩2 ∨ κ⇩1κ⇩2[Π]) → ∃β (β = Π)›
1226 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1227AOT_theorem "free-thms:4[2,1]": ‹([Π]κ⇩1κ⇩2 ∨ κ⇩1κ⇩2[Π]) → ∃β (β = κ⇩1)›
1228 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1229AOT_theorem "free-thms:4[2,2]": ‹([Π]κ⇩1κ⇩2 ∨ κ⇩1κ⇩2[Π]) → ∃β (β = κ⇩2)›
1230 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1231AOT_theorem "free-thms:4[3,rel]": ‹([Π]κ⇩1κ⇩2κ⇩3 ∨ κ⇩1κ⇩2κ⇩3[Π]) → ∃β (β = Π)›
1232 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1233AOT_theorem "free-thms:4[3,1]": ‹([Π]κ⇩1κ⇩2κ⇩3 ∨ κ⇩1κ⇩2κ⇩3[Π]) → ∃β (β = κ⇩1)›
1234 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1235AOT_theorem "free-thms:4[3,2]": ‹([Π]κ⇩1κ⇩2κ⇩3 ∨ κ⇩1κ⇩2κ⇩3[Π]) → ∃β (β = κ⇩2)›
1236 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1237AOT_theorem "free-thms:4[3,3]": ‹([Π]κ⇩1κ⇩2κ⇩3 ∨ κ⇩1κ⇩2κ⇩3[Π]) → ∃β (β = κ⇩3)›
1238 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1239AOT_theorem "free-thms:4[4,rel]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = Π)›
1240 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1241AOT_theorem "free-thms:4[4,1]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = κ⇩1)›
1242 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1243AOT_theorem "free-thms:4[4,2]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = κ⇩2)›
1244 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1245AOT_theorem "free-thms:4[4,3]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = κ⇩3)›
1246 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1247AOT_theorem "free-thms:4[4,4]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = κ⇩4)›
1248 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1249
1250AOT_theorem "ex:1:a": ‹∀α α↓›
1251 by (rule GEN) (fact "cqt:2[const_var]"[axiom_inst])
1252AOT_theorem "ex:1:b": ‹∀α∃β(β = α)›
1253 by (rule GEN) (fact "free-thms:3[const_var]")
1254
1255AOT_theorem "ex:2:a": ‹□α↓›
1256 by (rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1257AOT_theorem "ex:2:b": ‹□∃β(β = α)›
1258 by (rule RN) (fact "free-thms:3[const_var]")
1259
1260AOT_theorem "ex:3:a": ‹□∀α α↓›
1261 by (rule RN) (fact "ex:1:a")
1262AOT_theorem "ex:3:b": ‹□∀α∃β(β = α)›
1263 by (rule RN) (fact "ex:1:b")
1264
1265AOT_theorem "ex:4:a": ‹∀α □α↓›
1266 by (rule GEN; rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1267AOT_theorem "ex:4:b": ‹∀α□∃β(β = α)›
1268 by (rule GEN; rule RN) (fact "free-thms:3[const_var]")
1269
1270AOT_theorem "ex:5:a": ‹□∀α □α↓›
1271 by (rule RN) (simp add: "ex:4:a")
1272AOT_theorem "ex:5:b": ‹□∀α□∃β(β = α)›
1273 by (rule RN) (simp add: "ex:4:b")
1274
1275AOT_theorem "all-self=:1": ‹□∀α(α = α)›
1276 by (rule RN; rule GEN) (fact "id-eq:1")
1277AOT_theorem "all-self=:2": ‹∀α□(α = α)›
1278 by (rule GEN; rule RN) (fact "id-eq:1")
1279
1280AOT_theorem "id-nec:1": ‹α = β → □(α = β)›
1281proof(rule "→I")
1282 AOT_assume ‹α = β›
1283 moreover AOT_have ‹□(α = α)›
1284 by (rule RN) (fact "id-eq:1")
1285 ultimately AOT_show ‹□(α = β)› using "rule=E" by fast
1286qed
1287
1288AOT_theorem "id-nec:2": ‹τ = σ → □(τ = σ)›
1289proof(rule "→I")
1290 AOT_assume asm: ‹τ = σ›
1291 moreover AOT_have ‹τ↓›
1292 using calculation "t=t-proper:1" "→E" by blast
1293 moreover AOT_have ‹□(τ = τ)›
1294 using calculation "all-self=:2" "∀E"(1) by blast
1295 ultimately AOT_show ‹□(τ = σ)› using "rule=E" by fast
1296qed
1297
1298AOT_theorem "term-out:1": ‹φ{α} ≡ ∃β (β = α & φ{β})›
1299proof (rule "≡I"; rule "→I")
1300 AOT_assume asm: ‹φ{α}›
1301 AOT_show ‹∃β (β = α & φ{β})›
1302 by (rule "∃I"(2)[where β=α]; rule "&I")
1303 (auto simp: "id-eq:1" asm)
1304next
1305 AOT_assume 0: ‹∃β (β = α & φ{β})›
1306
1308 AOT_obtain β where ‹β = α & φ{β}› using "instantiation"[rotated, OF 0] by blast
1309 AOT_thus ‹φ{α}› using "&E" "rule=E" by blast
1310qed
1311
1312AOT_theorem "term-out:2": ‹τ↓ → (φ{τ} ≡ ∃α(α = τ & φ{α}))›
1313proof(rule "→I")
1314 AOT_assume ‹τ↓›
1315 moreover AOT_have ‹∀α (φ{α} ≡ ∃β (β = α & φ{β}))›
1316 by (rule GEN) (fact "term-out:1")
1317 ultimately AOT_show ‹φ{τ} ≡ ∃α(α = τ & φ{α})›
1318 using "∀E" by blast
1319qed
1320
1321
1322AOT_theorem "term-out:3": ‹(φ{α} & ∀β(φ{β} → β = α)) ≡ ∀β(φ{β} ≡ β = α)›
1323 apply (rule "≡I"; rule "→I")
1324 apply (frule "&E"(1)) apply (drule "&E"(2))
1325 apply (rule GEN; rule "≡I"; rule "→I")
1326 using "rule-ui:2[const_var]" "vdash-properties:5" apply blast
1327 apply (meson "rule=E" "id-eq:1")
1328 apply (rule "&I")
1329 using "id-eq:1" "≡E"(2) "rule-ui:3" apply blast
1330 apply (rule GEN; rule "→I")
1331 using "≡E"(1) "rule-ui:2[const_var]" by blast
1332
1333AOT_theorem "term-out:4": ‹(φ{β} & ∀α(φ{α} → α = β)) ≡ ∀α(φ{α} ≡ α = β)›
1334 using "term-out:3" .
1335
1336
1337AOT_define AOT_exists_unique :: ‹α ⇒ φ ⇒ φ›
1338 "uniqueness:1": ‹«AOT_exists_unique φ» ≡⇩d⇩f ∃α (φ{α} & ∀β (φ{β} → β = α))›
1339syntax "_AOT_exists_unique" :: ‹α ⇒ φ ⇒ φ› ("∃!_ _" [1,40])
1340AOT_syntax_print_translations
1341 "_AOT_exists_unique τ φ" <= "CONST AOT_exists_unique (_abs τ φ)"
1342syntax
1343 "_AOT_exists_unique_ellipse" :: ‹id_position ⇒ id_position ⇒ φ ⇒ φ› (‹∃!_...∃!_ _› [1,40])
1344parse_ast_translation‹[(\<^syntax_const>‹_AOT_exists_unique_ellipse›, fn ctx => fn [a,b,c] =>
1345 Ast.mk_appl (Ast.Constant "AOT_exists_unique") [parseEllipseList "_AOT_vars" ctx [a,b],c]),
1346(\<^syntax_const>‹_AOT_exists_unique›,AOT_restricted_binder \<^const_name>‹AOT_exists_unique› \<^const_syntax>‹AOT_conj›)]›
1347print_translation‹AOT_syntax_print_translations
1348 [AOT_preserve_binder_abs_tr' \<^const_syntax>‹AOT_exists_unique› \<^syntax_const>‹_AOT_exists_unique› (\<^syntax_const>‹_AOT_exists_unique_ellipse›, true) \<^const_name>‹AOT_conj›,
1349 AOT_binder_trans @{theory} @{binding "AOT_exists_unique_binder"} \<^syntax_const>‹_AOT_exists_unique›]
1350›
1351
1352
1353context AOT_meta_syntax
1354begin
1355notation AOT_exists_unique (binder "❙∃❙!" 20)
1356end
1357context AOT_no_meta_syntax
1358begin
1359no_notation AOT_exists_unique (binder "❙∃❙!" 20)
1360end
1361
1362AOT_theorem "uniqueness:2": ‹∃!α φ{α} ≡ ∃α∀β(φ{β} ≡ β = α)›
1363proof(rule "≡I"; rule "→I")
1364 AOT_assume ‹∃!α φ{α}›
1365 AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1366 using "uniqueness:1" "≡⇩d⇩fE" by blast
1367 then AOT_obtain α where ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1368 AOT_hence ‹∀β(φ{β} ≡ β = α)› using "term-out:3" "≡E" by blast
1369 AOT_thus ‹∃α∀β(φ{β} ≡ β = α)›
1370 using "∃I" by fast
1371next
1372 AOT_assume ‹∃α∀β(φ{β} ≡ β = α)›
1373 then AOT_obtain α where ‹∀β (φ{β} ≡ β = α)› using "instantiation"[rotated] by blast
1374 AOT_hence ‹φ{α} & ∀β (φ{β} → β = α)› using "term-out:3" "≡E" by blast
1375 AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1376 using "∃I" by fast
1377 AOT_thus ‹∃!α φ{α}›
1378 using "uniqueness:1" "≡⇩d⇩fI" by blast
1379qed
1380
1381AOT_theorem "uni-most": ‹∃!α φ{α} → ∀β∀γ((φ{β} & φ{γ}) → β = γ)›
1382proof(rule "→I"; rule GEN; rule GEN; rule "→I")
1383 fix β γ
1384 AOT_assume ‹∃!α φ{α}›
1385 AOT_hence ‹∃α∀β(φ{β} ≡ β = α)›
1386 using "uniqueness:2" "≡E" by blast
1387 then AOT_obtain α where ‹∀β(φ{β} ≡ β = α)›
1388 using "instantiation"[rotated] by blast
1389 moreover AOT_assume ‹φ{β} & φ{γ}›
1390 ultimately AOT_have ‹β = α› and ‹γ = α›
1391 using "∀E"(2) "&E" "≡E"(1,2) by blast+
1392 AOT_thus ‹β = γ›
1393 by (metis "rule=E" "id-eq:2" "→E")
1394qed
1395
1396AOT_theorem "nec-exist-!": ‹∀α(φ{α} → □φ{α}) → (∃!α φ{α} → ∃!α □φ{α})›
1397proof (rule "→I"; rule "→I")
1398 AOT_assume a: ‹∀α(φ{α} → □φ{α})›
1399 AOT_assume ‹∃!α φ{α}›
1400 AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))› using "uniqueness:1" "≡⇩d⇩fE" by blast
1401 then AOT_obtain α where ξ: ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1402 AOT_have ‹□φ{α}›
1403 using ξ a "&E" "∀E" "→E" by fast
1404 moreover AOT_have ‹∀β (□φ{β} → β = α)›
1405 apply (rule GEN; rule "→I")
1406 using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
1407 ultimately AOT_have ‹(□φ{α} & ∀β (□φ{β} → β = α))›
1408 using "&I" by blast
1409 AOT_thus ‹∃!α □φ{α}›
1410 using "uniqueness:1" "≡⇩d⇩fI" "∃I" by fast
1411qed
1412
1413AOT_theorem "act-cond": ‹❙𝒜(φ → ψ) → (❙𝒜φ → ❙𝒜ψ)›
1414 using "→I" "≡E"(1) "logic-actual-nec:2"[axiom_inst] by blast
1415
1416AOT_theorem "nec-imp-act": ‹□φ → ❙𝒜φ›
1417 by (metis "act-cond" "contraposition:1[2]" "≡E"(4) "qml:2"[THEN act_closure, axiom_inst] "qml-act:2"[axiom_inst] RAA(1) "→E" "→I")
1418
1419AOT_theorem "act-conj-act:1": ‹❙𝒜(❙𝒜φ → φ)›
1420 using "→I" "≡E"(2) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] by blast
1421
1422AOT_theorem "act-conj-act:2": ‹❙𝒜(φ → ❙𝒜φ)›
1423 by (metis "→I" "≡E"(2, 4) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] RAA(1))
1424
1425AOT_theorem "act-conj-act:3": ‹(❙𝒜φ & ❙𝒜ψ) → ❙𝒜(φ & ψ)›
1426proof -
1427 AOT_have ‹□(φ → (ψ → (φ & ψ)))›
1428 by (rule RN) (fact Adjunction)
1429 AOT_hence ‹❙𝒜(φ → (ψ → (φ & ψ)))›
1430 using "nec-imp-act" "→E" by blast
1431 AOT_hence ‹❙𝒜φ → ❙𝒜(ψ → (φ & ψ))›
1432 using "act-cond" "→E" by blast
1433 moreover AOT_have ‹❙𝒜(ψ → (φ & ψ)) → (❙𝒜ψ → ❙𝒜(φ & ψ))›
1434 by (fact "act-cond")
1435 ultimately AOT_have ‹❙𝒜φ → (❙𝒜ψ → ❙𝒜(φ & ψ))›
1436 using "→I" "→E" by metis
1437 AOT_thus ‹(❙𝒜φ & ❙𝒜ψ) → ❙𝒜(φ & ψ)›
1438 by (metis Importation "→E")
1439qed
1440
1441AOT_theorem "act-conj-act:4": ‹❙𝒜(❙𝒜φ ≡ φ)›
1442proof -
1443 AOT_have ‹(❙𝒜(❙𝒜φ → φ) & ❙𝒜(φ → ❙𝒜φ)) → ❙𝒜((❙𝒜φ → φ) & (φ → ❙𝒜φ))›
1444 by (fact "act-conj-act:3")
1445 moreover AOT_have ‹❙𝒜(❙𝒜φ → φ) & ❙𝒜(φ → ❙𝒜φ)›
1446 using "&I" "act-conj-act:1" "act-conj-act:2" by simp
1447 ultimately AOT_have ζ: ‹❙𝒜((❙𝒜φ → φ) & (φ → ❙𝒜φ))›
1448 using "→E" by blast
1449 AOT_have ‹❙𝒜(((❙𝒜φ → φ) & (φ → ❙𝒜φ)) → (❙𝒜φ ≡ φ))›
1450 using "conventions:3"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst] by blast
1451 AOT_hence ‹❙𝒜((❙𝒜φ → φ) & (φ → ❙𝒜φ)) → ❙𝒜(❙𝒜φ ≡ φ)›
1452 using "act-cond" "→E" by blast
1453 AOT_thus ‹❙𝒜(❙𝒜φ ≡ φ)› using ζ "→E" by blast
1454qed
1455
1456
1457inductive arbitrary_actualization for φ where
1458 ‹arbitrary_actualization φ «❙𝒜φ»›
1459| ‹arbitrary_actualization φ «❙𝒜ψ»› if ‹arbitrary_actualization φ ψ›
1460declare arbitrary_actualization.cases[AOT] arbitrary_actualization.induct[AOT]
1461 arbitrary_actualization.simps[AOT] arbitrary_actualization.intros[AOT]
1462syntax arbitrary_actualization :: ‹φ' ⇒ φ' ⇒ AOT_prop› ("ARBITRARY'_ACTUALIZATION'(_,_')")
1463
1464notepad
1465begin
1466 AOT_modally_strict {
1467 fix φ
1468 AOT_have ‹ARBITRARY_ACTUALIZATION(❙𝒜φ ≡ φ, ❙𝒜(❙𝒜φ ≡ φ))›
1469 using AOT_PLM.arbitrary_actualization.intros by metis
1470 AOT_have ‹ARBITRARY_ACTUALIZATION(❙𝒜φ ≡ φ, ❙𝒜❙𝒜(❙𝒜φ ≡ φ))›
1471 using AOT_PLM.arbitrary_actualization.intros by metis
1472 AOT_have ‹ARBITRARY_ACTUALIZATION(❙𝒜φ ≡ φ, ❙𝒜❙𝒜❙𝒜(❙𝒜φ ≡ φ))›
1473 using AOT_PLM.arbitrary_actualization.intros by metis
1474 }
1475end
1476
1477
1478AOT_theorem "closure-act:1": assumes ‹ARBITRARY_ACTUALIZATION(❙𝒜φ ≡ φ, ψ)› shows ‹ψ›
1479using assms proof(induct)
1480 case 1
1481 AOT_show ‹❙𝒜(❙𝒜φ ≡ φ)›
1482 by (simp add: "act-conj-act:4")
1483next
1484 case (2 ψ)
1485 AOT_thus ‹❙𝒜ψ›
1486 by (metis arbitrary_actualization.simps "≡E"(1) "logic-actual-nec:4"[axiom_inst])
1487qed
1488
1489AOT_theorem "closure-act:2": ‹∀α ❙𝒜(❙𝒜φ{α} ≡ φ{α})›
1490 by (simp add: "act-conj-act:4" "∀I")
1491
1492AOT_theorem "closure-act:3": ‹❙𝒜∀α ❙𝒜(❙𝒜φ{α} ≡ φ{α})›
1493 by (metis (no_types, lifting) "act-conj-act:4" "≡E"(1,2) "logic-actual-nec:3"[axiom_inst] "logic-actual-nec:4"[axiom_inst] "∀I")
1494
1495AOT_theorem "closure-act:4": ‹❙𝒜∀α⇩1...∀α⇩n ❙𝒜(❙𝒜φ{α⇩1...α⇩n} ≡ φ{α⇩1...α⇩n})›
1496 using "closure-act:3" .
1497
1498
1499AOT_theorem "RA[1]": assumes ‹❙⊢ φ› shows ‹❙⊢ ❙𝒜φ›
1500
1501 using "¬¬E" assms "≡E"(3) "logic-actual"[act_axiom_inst] "logic-actual-nec:1"[axiom_inst] "modus-tollens:2" by blast
1502AOT_theorem "RA[2]": assumes ‹❙⊢⇩□ φ› shows ‹❙𝒜φ›
1503
1504 using RN assms "nec-imp-act" "vdash-properties:5" by blast
1505AOT_theorem "RA[3]": assumes ‹Γ ❙⊢⇩□ φ› shows ‹❙𝒜Γ ❙⊢⇩□ ❙𝒜φ›
1506 using assms by (meson AOT_sem_act imageI)
1507
1508
1509AOT_act_theorem "ANeg:1": ‹¬❙𝒜φ ≡ ¬φ›
1510 by (simp add: "RA[1]" "contraposition:1[1]" "deduction-theorem" "≡I" "logic-actual"[act_axiom_inst])
1511
1512AOT_act_theorem "ANeg:2": ‹¬❙𝒜¬φ ≡ φ›
1513 using "ANeg:1" "≡I" "≡E"(5) "useful-tautologies:1" "useful-tautologies:2" by blast
1514
1515AOT_theorem "Act-Basic:1": ‹❙𝒜φ ∨ ❙𝒜¬φ›
1516 by (meson "∨I"(1,2) "≡E"(2) "logic-actual-nec:1"[axiom_inst] "raa-cor:1")
1517
1518AOT_theorem "Act-Basic:2": ‹❙𝒜(φ & ψ) ≡ (❙𝒜φ & ❙𝒜ψ)›
1519proof (rule "≡I"; rule "→I")
1520 AOT_assume ‹❙𝒜(φ & ψ)›
1521 moreover AOT_have ‹❙𝒜((φ & ψ) → φ)›
1522 by (simp add: "RA[2]" "Conjunction Simplification"(1))
1523 moreover AOT_have ‹❙𝒜((φ & ψ) → ψ)›
1524 by (simp add: "RA[2]" "Conjunction Simplification"(2))
1525 ultimately AOT_show ‹❙𝒜φ & ❙𝒜ψ›
1526 using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1527next
1528 AOT_assume ‹❙𝒜φ & ❙𝒜ψ›
1529 AOT_thus ‹❙𝒜(φ & ψ)›
1530 using "act-conj-act:3" "vdash-properties:6" by blast
1531qed
1532
1533AOT_theorem "Act-Basic:3": ‹❙𝒜(φ ≡ ψ) ≡ (❙𝒜(φ → ψ) & ❙𝒜(ψ → φ))›
1534proof (rule "≡I"; rule "→I")
1535 AOT_assume ‹❙𝒜(φ ≡ ψ)›
1536 moreover AOT_have ‹❙𝒜((φ ≡ ψ) → (φ → ψ))›
1537 by (simp add: "RA[2]" "deduction-theorem" "≡E"(1))
1538 moreover AOT_have ‹❙𝒜((φ ≡ ψ) → (ψ → φ))›
1539 by (simp add: "RA[2]" "deduction-theorem" "≡E"(2))
1540 ultimately AOT_show ‹❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)›
1541 using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1542next
1543 AOT_assume ‹❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)›
1544 AOT_hence ‹❙𝒜((φ → ψ) & (ψ → φ))›
1545 by (metis "act-conj-act:3" "vdash-properties:10")
1546 moreover AOT_have ‹❙𝒜(((φ → ψ) & (ψ → φ)) → (φ ≡ ψ))›
1547 by (simp add: "conventions:3" "RA[2]" "df-rules-formulas[2]" "vdash-properties:1[2]")
1548 ultimately AOT_show ‹❙𝒜(φ ≡ ψ)›
1549 using "act-cond"[THEN "→E", THEN "→E"] by metis
1550qed
1551
1552AOT_theorem "Act-Basic:4": ‹(❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)) ≡ (❙𝒜φ ≡ ❙𝒜ψ)›
1553proof (rule "≡I"; rule "→I")
1554 AOT_assume 0: ‹❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)›
1555 AOT_show ‹❙𝒜φ ≡ ❙𝒜ψ›
1556 using 0 "&E" "act-cond"[THEN "→E", THEN "→E"] "≡I" "→I" by metis
1557next
1558 AOT_assume ‹❙𝒜φ ≡ ❙𝒜ψ›
1559 AOT_thus ‹❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)›
1560 by (metis "→I" "logic-actual-nec:2"[axiom_inst] "≡E"(1,2) "&I")
1561qed
1562
1563AOT_theorem "Act-Basic:5": ‹❙𝒜(φ ≡ ψ) ≡ (❙𝒜φ ≡ ❙𝒜ψ)›
1564 using "Act-Basic:3" "Act-Basic:4" "≡E"(5) by blast
1565
1566AOT_theorem "Act-Basic:6": ‹❙𝒜φ ≡ □❙𝒜φ›
1567 by (simp add: "≡I" "qml:2"[axiom_inst] "qml-act:1"[axiom_inst])
1568
1569AOT_theorem "Act-Basic:7": ‹❙𝒜□φ → □❙𝒜φ›
1570 by (metis "Act-Basic:6" "→I" "→E" "≡E"(1,2) "nec-imp-act" "qml-act:2"[axiom_inst])
1571
1572AOT_theorem "Act-Basic:8": ‹□φ → □❙𝒜φ›
1573 using "Hypothetical Syllogism" "nec-imp-act" "qml-act:1"[axiom_inst] by blast
1574
1575AOT_theorem "Act-Basic:9": ‹❙𝒜(φ ∨ ψ) ≡ (❙𝒜φ ∨ ❙𝒜ψ)›
1576proof (rule "≡I"; rule "→I")
1577 AOT_assume ‹❙𝒜(φ ∨ ψ)›
1578 AOT_thus ‹❙𝒜φ ∨ ❙𝒜ψ›
1579 proof (rule "raa-cor:3")
1580 AOT_assume ‹¬(❙𝒜φ ∨ ❙𝒜ψ)›
1581 AOT_hence ‹¬❙𝒜φ & ¬❙𝒜ψ›
1582 by (metis "≡E"(1) "oth-class-taut:5:d")
1583 AOT_hence ‹❙𝒜¬φ & ❙𝒜¬ψ›
1584 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] "&E" "&I" by metis
1585 AOT_hence ‹❙𝒜(¬φ & ¬ψ)›
1586 using "≡E" "Act-Basic:2" by metis
1587 moreover AOT_have ‹❙𝒜((¬φ & ¬ψ) ≡ ¬(φ ∨ ψ))›
1588 using "RA[2]" "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:d" by blast
1589 moreover AOT_have ‹❙𝒜(¬φ & ¬ψ) ≡ ❙𝒜(¬(φ ∨ ψ))›
1590 using calculation(2) by (metis "Act-Basic:5" "≡E"(1))
1591 ultimately AOT_have ‹❙𝒜(¬(φ ∨ ψ))› using "≡E" by blast
1592 AOT_thus ‹¬❙𝒜(φ ∨ ψ)›
1593 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by auto
1594 qed
1595next
1596 AOT_assume ‹❙𝒜φ ∨ ❙𝒜ψ›
1597 AOT_thus ‹❙𝒜(φ ∨ ψ)›
1598 by (meson "RA[2]" "act-cond" "∨I"(1) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
1599qed
1600
1601AOT_theorem "Act-Basic:10": ‹❙𝒜∃α φ{α} ≡ ∃α ❙𝒜φ{α}›
1602proof -
1603 AOT_have θ: ‹¬❙𝒜∀α ¬φ{α} ≡ ¬∀α ❙𝒜¬φ{α}›
1604 by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1605 (metis "logic-actual-nec:3"[axiom_inst])
1606 AOT_have ξ: ‹¬∀α ❙𝒜¬φ{α} ≡ ¬∀α ¬❙𝒜φ{α}›
1607 by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1608 (rule "logic-actual-nec:1"[THEN universal_closure, axiom_inst, THEN "cqt-basic:3"[THEN "→E"]])
1609 AOT_have ‹❙𝒜(∃α φ{α}) ≡ ❙𝒜(¬∀α ¬φ{α})›
1610 using "conventions:4"[THEN "df-rules-formulas[1]", THEN act_closure, axiom_inst]
1611 "conventions:4"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst]
1612 "Act-Basic:4"[THEN "≡E"(1)] "&I" "Act-Basic:5"[THEN "≡E"(2)] by metis
1613 also AOT_have ‹… ≡ ¬❙𝒜∀α ¬φ{α}›
1614 by (simp add: "logic-actual-nec:1" "vdash-properties:1[2]")
1615 also AOT_have ‹… ≡ ¬∀α ❙𝒜 ¬φ{α}› using θ by blast
1616 also AOT_have ‹… ≡ ¬∀α ¬❙𝒜 φ{α}› using ξ by blast
1617 also AOT_have ‹… ≡ ∃α ❙𝒜 φ{α}›
1618 using "conventions:4"[THEN "≡Df"] by (metis "≡E"(6) "oth-class-taut:3:a")
1619 finally AOT_show ‹❙𝒜∃α φ{α} ≡ ∃α ❙𝒜φ{α}› .
1620qed
1621
1622
1623AOT_theorem "Act-Basic:11": ‹❙𝒜∀α(φ{α} ≡ ψ{α}) ≡ ∀α(❙𝒜φ{α} ≡ ❙𝒜ψ{α})›
1624proof(rule "≡I"; rule "→I")
1625 AOT_assume ‹❙𝒜∀α(φ{α} ≡ ψ{α})›
1626 AOT_hence ‹∀α❙𝒜(φ{α} ≡ ψ{α})›
1627 using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] by blast
1628 AOT_hence ‹❙𝒜(φ{α} ≡ ψ{α})› for α using "∀E" by blast
1629 AOT_hence ‹❙𝒜φ{α} ≡ ❙𝒜ψ{α}› for α by (metis "Act-Basic:5" "≡E"(1))
1630 AOT_thus ‹∀α(❙𝒜φ{α} ≡ ❙𝒜ψ{α})› by (rule "∀I")
1631next
1632 AOT_assume ‹∀α(❙𝒜φ{α} ≡ ❙𝒜ψ{α})›
1633 AOT_hence ‹❙𝒜φ{α} ≡ ❙𝒜ψ{α}› for α using "∀E" by blast
1634 AOT_hence ‹❙𝒜(φ{α} ≡ ψ{α})› for α by (metis "Act-Basic:5" "≡E"(2))
1635 AOT_hence ‹∀α ❙𝒜(φ{α} ≡ ψ{α})› by (rule "∀I")
1636 AOT_thus ‹❙𝒜∀α(φ{α} ≡ ψ{α})›
1637 using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
1638qed
1639
1640AOT_act_theorem "act-quant-uniq": ‹∀β(❙𝒜φ{β} ≡ β = α) ≡ ∀β(φ{β} ≡ β = α)›
1641proof(rule "≡I"; rule "→I")
1642 AOT_assume ‹∀β(❙𝒜φ{β} ≡ β = α)›
1643 AOT_hence ‹❙𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1644 AOT_hence ‹φ{β} ≡ β = α› for β
1645 using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1646 by metis
1647 AOT_thus ‹∀β(φ{β} ≡ β = α)› by (rule "∀I")
1648next
1649 AOT_assume ‹∀β(φ{β} ≡ β = α)›
1650 AOT_hence ‹φ{β} ≡ β = α› for β using "∀E" by blast
1651 AOT_hence ‹❙𝒜φ{β} ≡ β = α› for β
1652 using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1653 by metis
1654 AOT_thus ‹∀β(❙𝒜φ{β} ≡ β = α)› by (rule "∀I")
1655qed
1656
1657AOT_act_theorem "fund-cont-desc": ‹x = ❙ιx(φ{x}) ≡ ∀z(φ{z} ≡ z = x)›
1658 using descriptions[axiom_inst] "act-quant-uniq" "≡E"(5) by fast
1659
1660AOT_act_theorem hintikka: ‹x = ❙ιx(φ{x}) ≡ (φ{x} & ∀z (φ{z} → z = x))›
1661 using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" "fund-cont-desc" "≡E"(5) by blast
1662
1663
1664locale russel_axiom =
1665 fixes ψ
1666 assumes ψ_denotes_asm: "[v ⊨ ψ{κ}] ⟹ [v ⊨ κ↓]"
1667begin
1668AOT_act_theorem "russell-axiom": ‹ψ{❙ιx φ{x}} ≡ ∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1669proof -
1670 AOT_have b: ‹∀x (x = ❙ιx φ{x} ≡ (φ{x} & ∀z(φ{z} → z = x)))›
1671 using hintikka "∀I" by fast
1672 show ?thesis
1673 proof(rule "≡I"; rule "→I")
1674 AOT_assume c: ‹ψ{❙ιx φ{x}}›
1675 AOT_hence d: ‹❙ιx φ{x}↓› using ψ_denotes_asm by blast
1676 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1677 then AOT_obtain a where a_def: ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1678 moreover AOT_have ‹a = ❙ιx φ{x} ≡ (φ{a} & ∀z(φ{z} → z = a))› using b "∀E" by blast
1679 ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a)› using "≡E" by blast
1680 moreover AOT_have ‹ψ{a}›
1681 proof -
1682 AOT_have 1: ‹∀x∀y(x = y → y = x)›
1683 by (simp add: "id-eq:2" "universal-cor")
1684 AOT_have ‹a = ❙ιx φ{x} → ❙ιx φ{x} = a›
1685 by (rule "∀E"(1)[where τ="«❙ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1686 (auto simp: 1 d "universal-cor")
1687 AOT_thus ‹ψ{a}›
1688 using a_def c "rule=E" "→E" by blast
1689 qed
1690 ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a) & ψ{a}› by (rule "&I")
1691 AOT_thus ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})› by (rule "∃I")
1692 next
1693 AOT_assume ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1694 then AOT_obtain b where g: ‹φ{b} & ∀z(φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1695 AOT_hence h: ‹b = ❙ιx φ{x} ≡ (φ{b} & ∀z(φ{z} → z = b))› using b "∀E" by blast
1696 AOT_have ‹φ{b} & ∀z(φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1697 AOT_hence ‹b = ❙ιx φ{x}› using h "≡E" by blast
1698 AOT_thus ‹ψ{❙ιx φ{x}}› using j "rule=E" by blast
1699 qed
1700qed
1701end
1702
1703
1704
1706interpretation "russell-axiom[exe,1]": russel_axiom ‹λ κ . «[Π]κ»›
1707 by standard (metis "cqt:5:a[1]"[axiom_inst, THEN "→E"] "&E"(2))
1708interpretation "russell-axiom[exe,2,1,1]": russel_axiom ‹λ κ . «[Π]κκ'»›
1709 by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E")
1710interpretation "russell-axiom[exe,2,1,2]": russel_axiom ‹λ κ . «[Π]κ'κ»›
1711 by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1712interpretation "russell-axiom[exe,2,2]": russel_axiom ‹λ κ . «[Π]κκ»›
1713 by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1714interpretation "russell-axiom[exe,3,1,1]": russel_axiom ‹λ κ . «[Π]κκ'κ''»›
1715 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1716interpretation "russell-axiom[exe,3,1,2]": russel_axiom ‹λ κ . «[Π]κ'κκ''»›
1717 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1718interpretation "russell-axiom[exe,3,1,3]": russel_axiom ‹λ κ . «[Π]κ'κ''κ»›
1719 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1720interpretation "russell-axiom[exe,3,2,1]": russel_axiom ‹λ κ . «[Π]κκκ'»›
1721 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1722interpretation "russell-axiom[exe,3,2,2]": russel_axiom ‹λ κ . «[Π]κκ'κ»›
1723 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1724interpretation "russell-axiom[exe,3,2,3]": russel_axiom ‹λ κ . «[Π]κ'κκ»›
1725 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1726interpretation "russell-axiom[exe,3,3]": russel_axiom ‹λ κ . «[Π]κκκ»›
1727 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1728
1729interpretation "russell-axiom[enc,1]": russel_axiom ‹λ κ . «κ[Π]»›
1730 by standard (metis "cqt:5:b[1]"[axiom_inst, THEN "→E"] "&E"(2))
1731interpretation "russell-axiom[enc,2,1]": russel_axiom ‹λ κ . «κκ'[Π]»›
1732 by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E")
1733interpretation "russell-axiom[enc,2,2]": russel_axiom ‹λ κ . «κ'κ[Π]»›
1734 by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1735interpretation "russell-axiom[enc,2,3]": russel_axiom ‹λ κ . «κκ[Π]»›
1736 by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1737interpretation "russell-axiom[enc,3,1,1]": russel_axiom ‹λ κ . «κκ'κ''[Π]»›
1738 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1739interpretation "russell-axiom[enc,3,1,2]": russel_axiom ‹λ κ . «κ'κκ''[Π]»›
1740 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1741interpretation "russell-axiom[enc,3,1,3]": russel_axiom ‹λ κ . «κ'κ''κ[Π]»›
1742 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1743interpretation "russell-axiom[enc,3,2,1]": russel_axiom ‹λ κ . «κκκ'[Π]»›
1744 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1745interpretation "russell-axiom[enc,3,2,2]": russel_axiom ‹λ κ . «κκ'κ[Π]»›
1746 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1747interpretation "russell-axiom[enc,3,2,3]": russel_axiom ‹λ κ . «κ'κκ[Π]»›
1748 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1749interpretation "russell-axiom[enc,3,3]": russel_axiom ‹λ κ . «κκκ[Π]»›
1750 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1751
1752AOT_act_theorem "1-exists:1": ‹❙ιx φ{x}↓ ≡ ∃!x φ{x}›
1753proof(rule "≡I"; rule "→I")
1754 AOT_assume ‹❙ιx φ{x}↓›
1755 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1756 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1757 AOT_hence ‹φ{a} & ∀z (φ{z} → z = a)› using hintikka "≡E" by blast
1758 AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))› by (rule "∃I")
1759 AOT_thus ‹∃!x φ{x}› using "uniqueness:1"[THEN "≡⇩d⇩fI"] by blast
1760next
1761 AOT_assume ‹∃!x φ{x}›
1762 AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))›
1763 using "uniqueness:1"[THEN "≡⇩d⇩fE"] by blast
1764 then AOT_obtain b where ‹φ{b} & ∀z (φ{z} → z = b)› using "instantiation"[rotated] by blast
1765 AOT_hence ‹b = ❙ιx φ{x}› using hintikka "≡E" by blast
1766 AOT_thus ‹❙ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1767qed
1768
1769AOT_act_theorem "1-exists:2": ‹∃y(y=❙ιx φ{x}) ≡ ∃!x φ{x}›
1770 using "1-exists:1" "free-thms:1" "≡E"(6) by blast
1771
1772AOT_act_theorem "y-in:1": ‹x = ❙ιx φ{x} → φ{x}›
1773 using "&E"(1) "→I" hintikka "≡E"(1) by blast
1774
1775AOT_act_theorem "y-in:2": ‹z = ❙ιx φ{x} → φ{z}› using "y-in:1".
1776
1777AOT_act_theorem "y-in:3": ‹❙ιx φ{x}↓ → φ{❙ιx φ{x}}›
1778proof(rule "→I")
1779 AOT_assume ‹❙ιx φ{x}↓›
1780 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1781 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1782 moreover AOT_have ‹φ{a}› using calculation hintikka "≡E"(1) "&E" by blast
1783 ultimately AOT_show ‹φ{❙ιx φ{x}}› using "rule=E" by blast
1784qed
1785
1786AOT_act_theorem "y-in:4": ‹∃y (y = ❙ιx φ{x}) → φ{❙ιx φ{x}}›
1787 using "y-in:3"[THEN "→E"] "free-thms:1"[THEN "≡E"(2)] "→I" by blast
1788
1789
1790AOT_theorem "act-quant-nec": ‹∀β (❙𝒜φ{β} ≡ β = α) ≡ ∀β(❙𝒜❙𝒜φ{β} ≡ β = α)›
1791proof(rule "≡I"; rule "→I")
1792 AOT_assume ‹∀β (❙𝒜φ{β} ≡ β = α)›
1793 AOT_hence ‹❙𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1794 AOT_hence ‹❙𝒜❙𝒜φ{β} ≡ β = α› for β
1795 by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1796 AOT_thus ‹∀β(❙𝒜❙𝒜φ{β} ≡ β = α)›
1797 by (rule "∀I")
1798next
1799 AOT_assume ‹∀β(❙𝒜❙𝒜φ{β} ≡ β = α)›
1800 AOT_hence ‹❙𝒜❙𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1801 AOT_hence ‹❙𝒜φ{β} ≡ β = α› for β
1802 by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1803 AOT_thus ‹∀β (❙𝒜φ{β} ≡ β = α)›
1804 by (rule "∀I")
1805qed
1806
1807AOT_theorem "equi-desc-descA:1": ‹x = ❙ιx φ{x} ≡ x = ❙ιx(❙𝒜φ{x})›
1808proof -
1809 AOT_have ‹x = ❙ιx φ{x} ≡ ∀z (❙𝒜φ{z} ≡ z = x)› using descriptions[axiom_inst] by blast
1810 also AOT_have ‹... ≡ ∀z (❙𝒜❙𝒜φ{z} ≡ z = x)›
1811 proof(rule "≡I"; rule "→I"; rule "∀I")
1812 AOT_assume ‹∀z (❙𝒜φ{z} ≡ z = x)›
1813 AOT_hence ‹❙𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1814 AOT_thus ‹❙𝒜❙𝒜φ{a} ≡ a = x› for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1815 next
1816 AOT_assume ‹∀z (❙𝒜❙𝒜φ{z} ≡ z = x)›
1817 AOT_hence ‹❙𝒜❙𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1818 AOT_thus ‹❙𝒜φ{a} ≡ a = x› for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1819 qed
1820 also AOT_have ‹... ≡ x = ❙ιx(❙𝒜φ{x})›
1821 using "Commutativity of ≡"[THEN "≡E"(1)] descriptions[axiom_inst] by fast
1822 finally show ?thesis .
1823qed
1824
1825AOT_theorem "equi-desc-descA:2": ‹❙ιx φ{x}↓ → ❙ιx φ{x} = ❙ιx(❙𝒜φ{x})›
1826proof(rule "→I")
1827 AOT_assume ‹❙ιx φ{x}↓›
1828 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1829 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1830 moreover AOT_have ‹a = ❙ιx(❙𝒜φ{x})› using calculation "equi-desc-descA:1"[THEN "≡E"(1)] by blast
1831 ultimately AOT_show ‹❙ιx φ{x} = ❙ιx(❙𝒜φ{x})› using "rule=E" by fast
1832qed
1833
1834AOT_theorem "nec-hintikka-scheme": ‹x = ❙ιx φ{x} ≡ ❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x)›
1835proof -
1836 AOT_have ‹x = ❙ιx φ{x} ≡ ∀z(❙𝒜φ{z} ≡ z = x)› using descriptions[axiom_inst] by blast
1837 also AOT_have ‹… ≡ (❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x))›
1838 using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" by fast
1839 finally show ?thesis.
1840qed
1841
1842AOT_theorem "equiv-desc-eq:1": ‹❙𝒜∀x(φ{x} ≡ ψ{x}) → ∀x (x = ❙ιx φ{x} ≡ x = ❙ιx ψ{x})›
1843proof(rule "→I"; rule "∀I")
1844 fix β
1845 AOT_assume ‹❙𝒜∀x(φ{x} ≡ ψ{x})›
1846 AOT_hence ‹❙𝒜(φ{x} ≡ ψ{x})› for x using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] "∀E"(2) by blast
1847 AOT_hence 0: ‹❙𝒜φ{x} ≡ ❙𝒜ψ{x}› for x by (metis "Act-Basic:5" "≡E"(1))
1848 AOT_have ‹β = ❙ιx φ{x} ≡ ❙𝒜φ{β} & ∀z(❙𝒜φ{z} → z = β)› using "nec-hintikka-scheme" by blast
1849 also AOT_have ‹... ≡ ❙𝒜ψ{β} & ∀z(❙𝒜ψ{z} → z = β)›
1850 proof (rule "≡I"; rule "→I")
1851 AOT_assume 1: ‹❙𝒜φ{β} & ∀z(❙𝒜φ{z} → z = β)›
1852 AOT_hence ‹❙𝒜φ{z} → z = β› for z using "&E" "∀E" by blast
1853 AOT_hence ‹❙𝒜ψ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1854 AOT_hence ‹∀z(❙𝒜ψ{z} → z = β)› using "∀I" by fast
1855 moreover AOT_have ‹❙𝒜ψ{β}› using "&E" 0[THEN "≡E"(1)] 1 by blast
1856 ultimately AOT_show ‹❙𝒜ψ{β} & ∀z(❙𝒜ψ{z} → z = β)› using "&I" by blast
1857 next
1858 AOT_assume 1: ‹❙𝒜ψ{β} & ∀z(❙𝒜ψ{z} → z = β)›
1859 AOT_hence ‹❙𝒜ψ{z} → z = β› for z using "&E" "∀E" by blast
1860 AOT_hence ‹❙𝒜φ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1861 AOT_hence ‹∀z(❙𝒜φ{z} → z = β)› using "∀I" by fast
1862 moreover AOT_have ‹❙𝒜φ{β}› using "&E" 0[THEN "≡E"(2)] 1 by blast
1863 ultimately AOT_show ‹❙𝒜φ{β} & ∀z(❙𝒜φ{z} → z = β)› using "&I" by blast
1864 qed
1865 also AOT_have ‹... ≡ β = ❙ιx ψ{x}›
1866 using "Commutativity of ≡"[THEN "≡E"(1)] "nec-hintikka-scheme" by blast
1867 finally AOT_show ‹β = ❙ιx φ{x} ≡ β = ❙ιx ψ{x}› .
1868qed
1869
1870AOT_theorem "equiv-desc-eq:2": ‹❙ιx φ{x}↓ & ❙𝒜∀x(φ{x} ≡ ψ{x}) → ❙ιx φ{x} = ❙ιx ψ{x}›
1871proof(rule "→I")
1872 AOT_assume ‹❙ιx φ{x}↓ & ❙𝒜∀x(φ{x} ≡ ψ{x})›
1873 AOT_hence 0: ‹∃y (y = ❙ιx φ{x})› and
1874 1: ‹∀x (x = ❙ιx φ{x} ≡ x = ❙ιx ψ{x})›
1875 using "&E" "free-thms:1"[THEN "≡E"(1)] "equiv-desc-eq:1" "→E" by blast+
1876 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1877 moreover AOT_have ‹a = ❙ιx ψ{x}› using calculation 1 "∀E" "≡E"(1) by fast
1878 ultimately AOT_show ‹❙ιx φ{x} = ❙ιx ψ{x}›
1879 using "rule=E" by fast
1880qed
1881
1882AOT_theorem "equiv-desc-eq:3": ‹❙ιx φ{x}↓ & □∀x(φ{x} ≡ ψ{x}) → ❙ιx φ{x} = ❙ιx ψ{x}›
1883 using "→I" "equiv-desc-eq:2"[THEN "→E", OF "&I"] "&E" "nec-imp-act"[THEN "→E"] by metis
1884
1885
1886AOT_theorem "equiv-desc-eq:4": ‹❙ιx φ{x}↓ → □❙ιx φ{x}↓›
1887proof(rule "→I")
1888 AOT_assume ‹❙ιx φ{x}↓›
1889 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1890 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1891 AOT_thus ‹□❙ιx φ{x}↓›
1892 using "ex:2:a" "rule=E" by fast
1893qed
1894
1895AOT_theorem "equiv-desc-eq:5": ‹❙ιx φ{x}↓ → ∃y □(y = ❙ιx φ{x})›
1896proof(rule "→I")
1897 AOT_assume ‹❙ιx φ{x}↓›
1898 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1899 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1900 AOT_hence ‹□(a = ❙ιx φ{x})› by (metis "id-nec:2" "vdash-properties:10")
1901 AOT_thus ‹∃y □(y = ❙ιx φ{x})› by (rule "∃I")
1902qed
1903
1904AOT_act_theorem "equiv-desc-eq2:1": ‹∀x (φ{x} ≡ ψ{x}) → ∀x (x = ❙ιx φ{x} ≡ x = ❙ιx ψ{x})›
1905 using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:1"[THEN "→E"]
1906 "RA[1]" "deduction-theorem" by blast
1907
1908AOT_act_theorem "equiv-desc-eq2:2": ‹❙ιx φ{x}↓ & ∀x (φ{x} ≡ ψ{x}) → ❙ιx φ{x} = ❙ιx ψ{x}›
1909 using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:2"[THEN "→E", OF "&I"]
1910 "RA[1]" "deduction-theorem" "&E" by metis
1911
1912context russel_axiom
1913begin
1914AOT_theorem "nec-russell-axiom": ‹ψ{❙ιx φ{x}} ≡ ∃x(❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x) & ψ{x})›
1915proof -
1916 AOT_have b: ‹∀x (x = ❙ιx φ{x} ≡ (❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x)))›
1917 using "nec-hintikka-scheme" "∀I" by fast
1918 show ?thesis
1919 proof(rule "≡I"; rule "→I")
1920 AOT_assume c: ‹ψ{❙ιx φ{x}}›
1921 AOT_hence d: ‹❙ιx φ{x}↓› using ψ_denotes_asm by blast
1922 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1923 then AOT_obtain a where a_def: ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1924 moreover AOT_have ‹a = ❙ιx φ{x} ≡ (❙𝒜φ{a} & ∀z(❙𝒜φ{z} → z = a))› using b "∀E" by blast
1925 ultimately AOT_have ‹❙𝒜φ{a} & ∀z(❙𝒜φ{z} → z = a)› using "≡E" by blast
1926 moreover AOT_have ‹ψ{a}›
1927 proof -
1928 AOT_have 1: ‹∀x∀y(x = y → y = x)›
1929 by (simp add: "id-eq:2" "universal-cor")
1930 AOT_have ‹a = ❙ιx φ{x} → ❙ιx φ{x} = a›
1931 by (rule "∀E"(1)[where τ="«❙ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1932 (auto simp: d "universal-cor" 1)
1933 AOT_thus ‹ψ{a}›
1934 using a_def c "rule=E" "→E" by metis
1935 qed
1936 ultimately AOT_have ‹❙𝒜φ{a} & ∀z(❙𝒜φ{z} → z = a) & ψ{a}› by (rule "&I")
1937 AOT_thus ‹∃x(❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x) & ψ{x})› by (rule "∃I")
1938 next
1939 AOT_assume ‹∃x(❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x) & ψ{x})›
1940 then AOT_obtain b where g: ‹❙𝒜φ{b} & ∀z(❙𝒜φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1941 AOT_hence h: ‹b = ❙ιx φ{x} ≡ (❙𝒜φ{b} & ∀z(❙𝒜φ{z} → z = b))› using b "∀E" by blast
1942 AOT_have ‹❙𝒜φ{b} & ∀z(❙𝒜φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1943 AOT_hence ‹b = ❙ιx φ{x}› using h "≡E" by blast
1944 AOT_thus ‹ψ{❙ιx φ{x}}› using j "rule=E" by blast
1945 qed
1946qed
1947end
1948
1949AOT_theorem "actual-desc:1": ‹❙ιx φ{x}↓ ≡ ∃!x ❙𝒜φ{x}›
1950proof (rule "≡I"; rule "→I")
1951 AOT_assume ‹❙ιx φ{x}↓›
1952 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1953 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1954 moreover AOT_have ‹a = ❙ιx φ{x} ≡ ∀z(❙𝒜φ{z} ≡ z = a)›
1955 using descriptions[axiom_inst] by blast
1956 ultimately AOT_have ‹∀z(❙𝒜φ{z} ≡ z = a)›
1957 using "≡E" by blast
1958 AOT_hence ‹∃x∀z(❙𝒜φ{z} ≡ z = x)› by (rule "∃I")
1959 AOT_thus ‹∃!x ❙𝒜φ{x}›
1960 using "uniqueness:2"[THEN "≡E"(2)] by fast
1961next
1962 AOT_assume ‹∃!x ❙𝒜φ{x}›
1963 AOT_hence ‹∃x∀z(❙𝒜φ{z} ≡ z = x)›
1964 using "uniqueness:2"[THEN "≡E"(1)] by fast
1965 then AOT_obtain a where ‹∀z(❙𝒜φ{z} ≡ z = a)› using "instantiation"[rotated] by blast
1966 moreover AOT_have ‹a = ❙ιx φ{x} ≡ ∀z(❙𝒜φ{z} ≡ z = a)›
1967 using descriptions[axiom_inst] by blast
1968 ultimately AOT_have ‹a = ❙ιx φ{x}› using "≡E" by blast
1969 AOT_thus ‹❙ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1970qed
1971
1972AOT_theorem "actual-desc:2": ‹x = ❙ιx φ{x} → ❙𝒜φ{x}›
1973 using "&E"(1) "contraposition:1[2]" "≡E"(1) "nec-hintikka-scheme" "reductio-aa:2" "vdash-properties:9" by blast
1974
1975AOT_theorem "actual-desc:3": ‹z = ❙ιx φ{x} → ❙𝒜φ{z}›
1976 using "actual-desc:2".
1977
1978AOT_theorem "actual-desc:4": ‹❙ιx φ{x}↓ → ❙𝒜φ{❙ιx φ{x}}›
1979proof(rule "→I")
1980 AOT_assume ‹❙ιx φ{x}↓›
1981 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1982 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1983 AOT_thus ‹❙𝒜φ{❙ιx φ{x}}›
1984 using "actual-desc:2" "rule=E" "→E" by fast
1985qed
1986
1987
1988AOT_theorem "actual-desc:5": ‹❙ιx φ{x} = ❙ιx ψ{x} → ❙𝒜∀x(φ{x} ≡ ψ{x})›
1989proof(rule "→I")
1990 AOT_assume 0: ‹❙ιx φ{x} = ❙ιx ψ{x}›
1991 AOT_hence φ_down: ‹❙ιx φ{x}↓› and ψ_down: ‹❙ιx ψ{x}↓›
1992 using "t=t-proper:1" "t=t-proper:2" "vdash-properties:6" by blast+
1993 AOT_hence ‹∃y (y = ❙ιx φ{x})› and ‹∃y (y = ❙ιx ψ{x})› by (metis "rule=I:1" "existential:1")+
1994 then AOT_obtain a and b where a_eq: ‹a = ❙ιx φ{x}› and b_eq: ‹b = ❙ιx ψ{x}›
1995 using "instantiation"[rotated] by metis
1996
1997 AOT_have ‹∀α∀β (α = β → β = α)› by (rule "∀I"; rule "∀I"; rule "id-eq:2")
1998 AOT_hence ‹∀β (❙ιx φ{x} = β → β = ❙ιx φ{x})›
1999 using "∀E" φ_down by blast
2000 AOT_hence ‹❙ιx φ{x} = ❙ιx ψ{x} → ❙ιx ψ{x} = ❙ιx φ{x}›
2001 using "∀E" ψ_down by blast
2002 AOT_hence 1: ‹❙ιx ψ{x} = ❙ιx φ{x}› using 0
2003 "→E" by blast
2004
2005 AOT_have ‹❙𝒜φ{x} ≡ ❙𝒜ψ{x}› for x
2006 proof(rule "≡I"; rule "→I")
2007 AOT_assume ‹❙𝒜φ{x}›
2008 moreover AOT_have ‹❙𝒜φ{x} → x = a› for x
2009 using "nec-hintikka-scheme"[THEN "≡E"(1), OF a_eq, THEN "&E"(2)] "∀E" by blast
2010 ultimately AOT_have ‹x = a› using "→E" by blast
2011 AOT_hence ‹x = ❙ιx φ{x}› using a_eq "rule=E" by blast
2012 AOT_hence ‹x = ❙ιx ψ{x}› using 0 "rule=E" by blast
2013 AOT_thus ‹❙𝒜ψ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2014 next
2015 AOT_assume ‹❙𝒜ψ{x}›
2016 moreover AOT_have ‹❙𝒜ψ{x} → x = b› for x
2017 using "nec-hintikka-scheme"[THEN "≡E"(1), OF b_eq, THEN "&E"(2)] "∀E" by blast
2018 ultimately AOT_have ‹x = b› using "→E" by blast
2019 AOT_hence ‹x = ❙ιx ψ{x}› using b_eq "rule=E" by blast
2020 AOT_hence ‹x = ❙ιx φ{x}› using 1 "rule=E" by blast
2021 AOT_thus ‹❙𝒜φ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2022 qed
2023 AOT_hence ‹❙𝒜(φ{x} ≡ ψ{x})› for x by (metis "Act-Basic:5" "≡E"(2))
2024 AOT_hence ‹∀x ❙𝒜(φ{x} ≡ ψ{x})› by (rule "∀I")
2025 AOT_thus ‹❙𝒜∀x (φ{x} ≡ ψ{x})›
2026 using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
2027qed
2028
2029AOT_theorem "!box-desc:1": ‹∃!x □φ{x} → ∀y (y = ❙ιx φ{x} → φ{y})›
2030proof(rule "→I")
2031 AOT_assume ‹∃!x □φ{x}›
2032 AOT_hence ζ: ‹∃x (□φ{x} & ∀z (□φ{z} → z = x))›
2033 using "uniqueness:1"[THEN "≡⇩d⇩fE"] by blast
2034 then AOT_obtain b where θ: ‹□φ{b} & ∀z (□φ{z} → z = b)› using "instantiation"[rotated] by blast
2035 AOT_show ‹∀y (y = ❙ιx φ{x} → φ{y})›
2036 proof(rule GEN; rule "→I")
2037 fix y
2038 AOT_assume ‹y = ❙ιx φ{x}›
2039 AOT_hence ‹❙𝒜φ{y} & ∀z (❙𝒜φ{z} → z = y)› using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
2040 AOT_hence ‹❙𝒜φ{b} → b = y› using "&E" "∀E" by blast
2041 moreover AOT_have ‹❙𝒜φ{b}› using θ[THEN "&E"(1)] by (metis "nec-imp-act" "→E")
2042 ultimately AOT_have ‹b = y› using "→E" by blast
2043 moreover AOT_have ‹φ{b}› using θ[THEN "&E"(1)] by (metis "qml:2"[axiom_inst] "→E")
2044 ultimately AOT_show ‹φ{y}› using "rule=E" by blast
2045 qed
2046qed
2047
2048AOT_theorem "!box-desc:2": ‹∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ∀y (y = ❙ιx φ{x} → φ{y}))›
2049proof(rule "→I"; rule "→I")
2050 AOT_assume ‹∀x (φ{x} → □φ{x})›
2051 moreover AOT_assume ‹∃!x φ{x}›
2052 ultimately AOT_have ‹∃!x □φ{x}›
2053 using "nec-exist-!"[THEN "→E", THEN "→E"] by blast
2054 AOT_thus ‹∀y (y = ❙ιx φ{x} → φ{y})›
2055 using "!box-desc:1" "→E" by blast
2056qed
2057
2058AOT_theorem "dr-alphabetic-thm": ‹❙ιν φ{ν}↓ → ❙ιν φ{ν} = ❙ιμ φ{μ}›
2059 by (simp add: "rule=I:1" "→I")
2060
2061AOT_theorem "RM:1[prem]": assumes ‹Γ ❙⊢⇩□ φ → ψ› shows ‹□Γ ❙⊢⇩□ □φ → □ψ›
2062proof -
2063 AOT_have ‹□Γ ❙⊢⇩□ □(φ → ψ)› using "RN[prem]" assms by blast
2064 AOT_thus ‹□Γ ❙⊢⇩□ □φ → □ψ› by (metis "qml:1"[axiom_inst] "→E")
2065qed
2066
2067AOT_theorem "RM:1": assumes ‹❙⊢⇩□ φ → ψ› shows ‹❙⊢⇩□ □φ → □ψ›
2068 using "RM:1[prem]" assms by blast
2069
2070lemmas RM = "RM:1"
2071
2072AOT_theorem "RM:2[prem]": assumes ‹Γ ❙⊢⇩□ φ → ψ› shows ‹□Γ ❙⊢⇩□ ◇φ → ◇ψ›
2073proof -
2074 AOT_have ‹Γ ❙⊢⇩□ ¬ψ → ¬φ› using assms
2075 by (simp add: "contraposition:1[1]")
2076 AOT_hence ‹□Γ ❙⊢⇩□ □¬ψ → □¬φ› using "RM:1[prem]" by blast
2077 AOT_thus ‹□Γ ❙⊢⇩□ ◇φ → ◇ψ›
2078 by (meson "≡⇩d⇩fE" "≡⇩d⇩fI" "conventions:5" "deduction-theorem" "modus-tollens:1")
2079qed
2080
2081AOT_theorem "RM:2": assumes ‹❙⊢⇩□ φ → ψ› shows ‹❙⊢⇩□ ◇φ → ◇ψ›
2082 using "RM:2[prem]" assms by blast
2083
2084lemmas "RM◇" = "RM:2"
2085
2086AOT_theorem "RM:3[prem]": assumes ‹Γ ❙⊢⇩□ φ ≡ ψ› shows ‹□Γ ❙⊢⇩□ □φ ≡ □ψ›
2087proof -
2088 AOT_have ‹Γ ❙⊢⇩□ φ → ψ› and ‹Γ ❙⊢⇩□ ψ → φ› using assms "≡E" "→I" by metis+
2089 AOT_hence ‹□Γ ❙⊢⇩□ □φ → □ψ› and ‹□Γ ❙⊢⇩□ □ψ → □φ› using "RM:1[prem]" by metis+
2090 AOT_thus ‹□Γ ❙⊢⇩□ □φ ≡ □ψ›
2091 by (simp add: "≡I")
2092qed
2093
2094AOT_theorem "RM:3": assumes ‹❙⊢⇩□ φ ≡ ψ› shows ‹❙⊢⇩□ □φ ≡ □ψ›
2095 using "RM:3[prem]" assms by blast
2096
2097lemmas RE = "RM:3"
2098
2099AOT_theorem "RM:4[prem]": assumes ‹Γ ❙⊢⇩□ φ ≡ ψ› shows ‹□Γ ❙⊢⇩□ ◇φ ≡ ◇ψ›
2100proof -
2101 AOT_have ‹Γ ❙⊢⇩□ φ → ψ› and ‹Γ ❙⊢⇩□ ψ → φ› using assms "≡E" "→I" by metis+
2102 AOT_hence ‹□Γ ❙⊢⇩□ ◇φ → ◇ψ› and ‹□Γ ❙⊢⇩□ ◇ψ → ◇φ› using "RM:2[prem]" by metis+
2103 AOT_thus ‹□Γ ❙⊢⇩□ ◇φ ≡ ◇ψ› by (simp add: "≡I")
2104qed
2105
2106AOT_theorem "RM:4": assumes ‹❙⊢⇩□ φ ≡ ψ› shows ‹❙⊢⇩□ ◇φ ≡ ◇ψ›
2107 using "RM:4[prem]" assms by blast
2108
2109lemmas "RE◇" = "RM:4"
2110
2111AOT_theorem "KBasic:1": ‹□φ → □(ψ → φ)›
2112 by (simp add: RM "pl:1"[axiom_inst])
2113
2114AOT_theorem "KBasic:2": ‹□¬φ → □(φ → ψ)›
2115 by (simp add: RM "useful-tautologies:3")
2116
2117AOT_theorem "KBasic:3": ‹□(φ & ψ) ≡ (□φ & □ψ)›
2118proof (rule "≡I"; rule "→I")
2119 AOT_assume ‹□(φ & ψ)›
2120 AOT_thus ‹□φ & □ψ›
2121 by (meson RM "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2122next
2123 AOT_have ‹□φ → □(ψ → (φ & ψ))› by (simp add: "RM:1" Adjunction)
2124 AOT_hence ‹□φ → (□ψ → □(φ & ψ))› by (metis "Hypothetical Syllogism" "qml:1"[axiom_inst])
2125 moreover AOT_assume ‹□φ & □ψ›
2126 ultimately AOT_show ‹□(φ & ψ)›
2127 using "→E" "&E" by blast
2128qed
2129
2130AOT_theorem "KBasic:4": ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2131proof -
2132 AOT_have θ: ‹□((φ → ψ) & (ψ → φ)) ≡ (□(φ → ψ) & □(ψ → φ))›
2133 by (fact "KBasic:3")
2134 AOT_modally_strict {
2135 AOT_have ‹(φ ≡ ψ) ≡ ((φ → ψ) & (ψ → φ))›
2136 by (fact "conventions:3"[THEN "≡Df"])
2137 }
2138 AOT_hence ξ: ‹□(φ ≡ ψ) ≡ □((φ → ψ) & (ψ → φ))›
2139 by (rule RE)
2140 with ξ and θ AOT_show ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2141 using "≡E"(5) by blast
2142qed
2143
2144AOT_theorem "KBasic:5": ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2145proof -
2146 AOT_have ‹□(φ → ψ) → (□φ → □ψ)›
2147 by (fact "qml:1"[axiom_inst])
2148 moreover AOT_have ‹□(ψ → φ) → (□ψ → □φ)›
2149 by (fact "qml:1"[axiom_inst])
2150 ultimately AOT_have ‹(□(φ → ψ) & □(ψ → φ)) → ((□φ → □ψ) & (□ψ → □φ))›
2151 by (metis "&I" MP "Double Composition")
2152 moreover AOT_have ‹((□φ → □ψ) & (□ψ → □φ)) → (□φ ≡ □ψ)›
2153 using "conventions:3"[THEN "≡⇩d⇩fI"] "→I" by blast
2154 ultimately AOT_show ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2155 by (metis "Hypothetical Syllogism")
2156qed
2157
2158AOT_theorem "KBasic:6": ‹□(φ≡ ψ) → (□φ ≡ □ψ)›
2159 using "KBasic:4" "KBasic:5" "deduction-theorem" "≡E"(1) "vdash-properties:10" by blast
2160AOT_theorem "KBasic:7": ‹((□φ & □ψ) ∨ (□¬φ & □¬ψ)) → □(φ ≡ ψ)›
2161proof (rule "→I"; drule "∨E"(1); (rule "→I")?)
2162 AOT_assume ‹□φ & □ψ›
2163 AOT_hence ‹□φ› and ‹□ψ› using "&E" by blast+
2164 AOT_hence ‹□(φ → ψ)› and ‹□(ψ → φ)› using "KBasic:1" "→E" by blast+
2165 AOT_hence ‹□(φ → ψ) & □(ψ → φ)› using "&I" by blast
2166 AOT_thus ‹□(φ ≡ ψ)› by (metis "KBasic:4" "≡E"(2))
2167next
2168 AOT_assume ‹□¬φ & □¬ψ›
2169 AOT_hence 0: ‹□(¬φ & ¬ψ)› using "KBasic:3"[THEN "≡E"(2)] by blast
2170 AOT_modally_strict {
2171 AOT_have ‹(¬φ & ¬ψ) → (φ ≡ ψ)›
2172 by (metis "&E"(1) "&E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
2173 }
2174 AOT_hence ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2175 by (rule RM)
2176 AOT_thus ‹□(φ ≡ ψ)› using 0 "→E" by blast
2177qed(auto)
2178
2179AOT_theorem "KBasic:8": ‹□(φ & ψ) → □(φ ≡ ψ)›
2180 by (meson "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I")
2181AOT_theorem "KBasic:9": ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2182 by (metis "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I" "raa-cor:4")
2183AOT_theorem "KBasic:10": ‹□φ ≡ □¬¬φ›
2184 by (simp add: "RM:3" "oth-class-taut:3:b")
2185AOT_theorem "KBasic:11": ‹¬□φ ≡ ◇¬φ›
2186proof (rule "≡I"; rule "→I")
2187 AOT_show ‹◇¬φ› if ‹¬□φ›
2188 using that "≡⇩d⇩fI" "conventions:5" "KBasic:10" "≡E"(3) by blast
2189next
2190 AOT_show ‹¬□φ› if ‹◇¬φ›
2191 using "≡⇩d⇩fE" "conventions:5" "KBasic:10" "≡E"(4) that by blast
2192qed
2193AOT_theorem "KBasic:12": ‹□φ ≡ ¬◇¬φ›
2194proof (rule "≡I"; rule "→I")
2195 AOT_show ‹¬◇¬φ› if ‹□φ›
2196 using "¬¬I" "KBasic:11" "≡E"(3) that by blast
2197next
2198 AOT_show ‹□φ› if ‹¬◇¬φ›
2199 using "KBasic:11" "≡E"(1) "reductio-aa:1" that by blast
2200qed
2201AOT_theorem "KBasic:13": ‹□(φ → ψ) → (◇φ → ◇ψ)›
2202proof -
2203 AOT_have ‹φ → ψ ❙⊢⇩□ φ → ψ› by blast
2204 AOT_hence ‹□(φ → ψ) ❙⊢⇩□ ◇φ → ◇ψ›
2205 using "RM:2[prem]" by blast
2206 AOT_thus ‹□(φ → ψ) → (◇φ → ◇ψ)› using "→I" by blast
2207qed
2208lemmas "K◇" = "KBasic:13"
2209AOT_theorem "KBasic:14": ‹◇□φ ≡ ¬□◇¬φ›
2210 by (meson "RE◇" "KBasic:11" "KBasic:12" "≡E"(6) "oth-class-taut:3:a")
2211AOT_theorem "KBasic:15": ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2212proof -
2213 AOT_modally_strict {
2214 AOT_have ‹φ → (φ ∨ ψ)› and ‹ψ → (φ ∨ ψ)›
2215 by (auto simp: "Disjunction Addition"(1) "Disjunction Addition"(2))
2216 }
2217 AOT_hence ‹□φ → □(φ ∨ ψ)› and ‹□ψ → □(φ ∨ ψ)›
2218 using RM by blast+
2219 AOT_thus ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2220 by (metis "∨E"(1) "deduction-theorem")
2221qed
2222
2223AOT_theorem "KBasic:16": ‹(□φ & ◇ψ) → ◇(φ & ψ)›
2224 by (meson "KBasic:13" "RM:1" Adjunction "Hypothetical Syllogism" Importation "vdash-properties:6")
2225
2226AOT_theorem "rule-sub-lem:1:a":
2227 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2228 shows ‹❙⊢⇩□ ¬ψ ≡ ¬χ›
2229 using "qml:2"[axiom_inst, THEN "→E", OF assms]
2230 "≡E"(1) "oth-class-taut:4:b" by blast
2231
2232AOT_theorem "rule-sub-lem:1:b":
2233 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2234 shows ‹❙⊢⇩□ (ψ → Θ) ≡ (χ → Θ)›
2235 using "qml:2"[axiom_inst, THEN "→E", OF assms]
2236 using "oth-class-taut:4:c" "vdash-properties:6" by blast
2237
2238AOT_theorem "rule-sub-lem:1:c":
2239 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2240 shows ‹❙⊢⇩□ (Θ → ψ) ≡ (Θ → χ)›
2241 using "qml:2"[axiom_inst, THEN "→E", OF assms]
2242 using "oth-class-taut:4:d" "vdash-properties:6" by blast
2243
2244AOT_theorem "rule-sub-lem:1:d":
2245 assumes ‹for arbitrary α: ❙⊢⇩□ □(ψ{α} ≡ χ{α})›
2246 shows ‹❙⊢⇩□ ∀α ψ{α} ≡ ∀α χ{α}›
2247proof -
2248 AOT_modally_strict {
2249 AOT_have ‹∀α (ψ{α} ≡ χ{α})›
2250 using "qml:2"[axiom_inst, THEN "→E", OF assms] "∀I" by fast
2251 AOT_hence 0: ‹ψ{α} ≡ χ{α}› for α using "∀E" by blast
2252 AOT_show ‹∀α ψ{α} ≡ ∀α χ{α}›
2253 proof (rule "≡I"; rule "→I")
2254 AOT_assume ‹∀α ψ{α}›
2255 AOT_hence ‹ψ{α}› for α using "∀E" by blast
2256 AOT_hence ‹χ{α}› for α using 0 "≡E" by blast
2257 AOT_thus ‹∀α χ{α}› by (rule "∀I")
2258 next
2259 AOT_assume ‹∀α χ{α}›
2260 AOT_hence ‹χ{α}› for α using "∀E" by blast
2261 AOT_hence ‹ψ{α}› for α using 0 "≡E" by blast
2262 AOT_thus ‹∀α ψ{α}› by (rule "∀I")
2263 qed
2264 }
2265qed
2266
2267AOT_theorem "rule-sub-lem:1:e":
2268 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2269 shows ‹❙⊢⇩□ [λ ψ] ≡ [λ χ]›
2270 using "qml:2"[axiom_inst, THEN "→E", OF assms]
2271 using "≡E"(1) "propositions-lemma:6" by blast
2272
2273AOT_theorem "rule-sub-lem:1:f":
2274 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2275 shows ‹❙⊢⇩□ ❙𝒜ψ ≡ ❙𝒜χ›
2276 using "qml:2"[axiom_inst, THEN "→E", OF assms, THEN "RA[2]"]
2277 by (metis "Act-Basic:5" "≡E"(1))
2278
2279AOT_theorem "rule-sub-lem:1:g":
2280 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2281 shows ‹❙⊢⇩□ □ψ ≡ □χ›
2282 using "KBasic:6" assms "vdash-properties:6" by blast
2283
2284text‹Note that instead of deriving @{text "rule-sub-lem:2"}, @{text "rule-sub-lem:3"}, @{text "rule-sub-lem:4"},
2285 and @{text "rule-sub-nec"}, we construct substitution methods instead.›
2286
2287class AOT_subst =
2288 fixes AOT_subst :: "('a ⇒ 𝗈) ⇒ bool"
2289 and AOT_subst_cond :: "'a ⇒ 'a ⇒ bool"
2290 assumes AOT_subst: "AOT_subst φ ⟹ AOT_subst_cond ψ χ ⟹ [v ⊨ «φ ψ» ≡ «φ χ»]"
2291
2292named_theorems AOT_substI
2293
2294instantiation 𝗈 :: AOT_subst
2295begin
2296
2297inductive AOT_subst_𝗈 where
2298 AOT_subst_𝗈_id[AOT_substI]: "AOT_subst_𝗈 (λφ. φ)"
2299| AOT_subst_𝗈_const[AOT_substI]: "AOT_subst_𝗈 (λφ. ψ)"
2300| AOT_subst_𝗈_not[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «¬Θ{φ}»)"
2301| AOT_subst_𝗈_imp[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 (λ φ. «Θ{φ} → Ξ{φ}»)"
2302| AOT_subst_𝗈_lambda0[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. (AOT_lambda0 (Θ φ)))"
2303| AOT_subst_𝗈_act[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «❙𝒜Θ{φ}»)"
2304| AOT_subst_𝗈_box[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «□Θ{φ}»)"
2305| AOT_subst_𝗈_by_def[AOT_substI]: "(⋀ ψ . AOT_model_equiv_def (Θ ψ) (Ξ ψ)) ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 Θ"
2306
2307definition AOT_subst_cond_𝗈 where "AOT_subst_cond_𝗈 ≡ λ ψ χ . ∀ v . [v ⊨ ψ ≡ χ]"
2308
2309instance
2310proof
2311 fix ψ χ :: 𝗈 and φ :: ‹𝗈 ⇒ 𝗈›
2312 assume cond: ‹AOT_subst_cond ψ χ›
2313 assume ‹AOT_subst φ›
2314 moreover AOT_have ‹❙⊢⇩□ ψ ≡ χ› using cond unfolding AOT_subst_cond_𝗈_def by blast
2315 ultimately AOT_show ‹❙⊢⇩□ φ{ψ} ≡ φ{χ}›
2316 proof (induct arbitrary: ψ χ)
2317 case AOT_subst_𝗈_id
2318 thus ?case using "≡E"(2) "oth-class-taut:4:b" "rule-sub-lem:1:a" by blast
2319 next
2320 case (AOT_subst_𝗈_const ψ)
2321 thus ?case by (simp add: "oth-class-taut:3:a")
2322 next
2323 case (AOT_subst_𝗈_not Θ)
2324 thus ?case by (simp add: RN "rule-sub-lem:1:a")
2325 next
2326 case (AOT_subst_𝗈_imp Θ Ξ)
2327 thus ?case by (meson RN "≡E"(5) "rule-sub-lem:1:b" "rule-sub-lem:1:c")
2328 next
2329 case (AOT_subst_𝗈_lambda0 Θ)
2330 thus ?case by (simp add: RN "rule-sub-lem:1:e")
2331 next
2332 case (AOT_subst_𝗈_act Θ)
2333 thus ?case by (simp add: RN "rule-sub-lem:1:f")
2334 next
2335 case (AOT_subst_𝗈_box Θ)
2336 thus ?case by (simp add: RN "rule-sub-lem:1:g")
2337 next
2338 case (AOT_subst_𝗈_by_def Θ Ξ)
2339 AOT_modally_strict {
2340 AOT_have ‹Ξ{ψ} ≡ Ξ{χ}› using AOT_subst_𝗈_by_def by simp
2341 AOT_thus ‹Θ{ψ} ≡ Θ{χ}›
2342 using "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ ψ] "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ χ]
2343 by (metis "≡E"(6) "oth-class-taut:3:a")
2344 }
2345 qed
2346qed
2347end
2348
2349instantiation "fun" :: (AOT_Term_id_2, AOT_subst) AOT_subst
2350begin
2351
2352definition AOT_subst_cond_fun :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b) ⇒ bool" where
2353 "AOT_subst_cond_fun ≡ λ φ ψ . ∀ α . AOT_subst_cond (φ (AOT_term_of_var α)) (ψ (AOT_term_of_var α))"
2354
2355inductive AOT_subst_fun :: "(('a ⇒ 'b) ⇒ 𝗈) ⇒ bool" where
2356 AOT_subst_fun_const[AOT_substI]: "AOT_subst_fun (λφ. ψ)"
2357| AOT_subst_fun_id[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. Ψ (φ (AOT_term_of_var x)))"
2358| AOT_subst_fun_all[AOT_substI]: "AOT_subst Ψ ⟹ (⋀ α . AOT_subst_fun (Θ (AOT_term_of_var α))) ⟹ AOT_subst_fun (λφ :: 'a ⇒ 'b. Ψ «∀α «Θ (α::'a) φ»»)"
2359| AOT_subst_fun_not[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. «¬«Ψ φ»»)"
2360| AOT_subst_fun_imp[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst Θ ⟹ AOT_subst_fun (λφ. ««Ψ φ» → «Θ φ»»)"
2361| AOT_subst_fun_lambda0[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. (AOT_lambda0 (Θ φ)))"
2362| AOT_subst_fun_act[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «❙𝒜«Θ φ»»)"
2363| AOT_subst_fun_box[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «□«Θ φ»»)"
2364| AOT_subst_fun_def[AOT_substI]: "(⋀ φ . AOT_model_equiv_def (Θ φ) (Ψ φ)) ⟹ AOT_subst_fun Ψ ⟹ AOT_subst_fun Θ"
2365
2366instance proof
2367 fix ψ χ :: "'a ⇒ 'b" and φ :: ‹('a ⇒ 'b) ⇒ 𝗈›
2368 assume ‹AOT_subst φ›
2369 moreover assume cond: ‹AOT_subst_cond ψ χ›
2370 ultimately AOT_show ‹❙⊢⇩□ «φ ψ» ≡ «φ χ»›
2371 proof(induct)
2372 case (AOT_subst_fun_const ψ)
2373 then show ?case by (simp add: "oth-class-taut:3:a")
2374 next
2375 case (AOT_subst_fun_id Ψ x)
2376 then show ?case by (simp add: AOT_subst AOT_subst_cond_fun_def)
2377 next
2378 case (AOT_subst_fun_all Ψ Θ)
2379 AOT_have ‹❙⊢⇩□ □(Θ{α, «ψ»} ≡ Θ{α, «χ»})› for α
2380 using AOT_subst_fun_all.hyps(3) AOT_subst_fun_all.prems RN by presburger
2381 thus ?case using AOT_subst[OF AOT_subst_fun_all(1)]
2382 by (simp add: RN "rule-sub-lem:1:d" AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def)
2383 next
2384 case (AOT_subst_fun_not Ψ)
2385 then show ?case by (simp add: RN "rule-sub-lem:1:a")
2386 next
2387 case (AOT_subst_fun_imp Ψ Θ)
2388 then show ?case
2389 unfolding AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def
2390 by (meson "≡E"(5) "oth-class-taut:4:c" "oth-class-taut:4:d" "vdash-properties:6")
2391 next
2392 case (AOT_subst_fun_lambda0 Θ)
2393 then show ?case by (simp add: RN "rule-sub-lem:1:e")
2394 next
2395 case (AOT_subst_fun_act Θ)
2396 then show ?case by (simp add: RN "rule-sub-lem:1:f")
2397 next
2398 case (AOT_subst_fun_box Θ)
2399 then show ?case by (simp add: RN "rule-sub-lem:1:g")
2400 next
2401 case (AOT_subst_fun_def Θ Ψ)
2402 then show ?case
2403 by (meson "df-rules-formulas[3]" "df-rules-formulas[4]" "≡I" "≡E"(5))
2404 qed
2405qed
2406end
2407
2408method_setup AOT_defI =
2409‹Scan.lift (Scan.succeed (fn ctxt => (Method.CONTEXT_METHOD (fn thms => (Context_Tactic.CONTEXT_SUBGOAL (fn (trm,int) =>
2410Context_Tactic.CONTEXT_TACTIC (
2411let
2412fun findHeadConst (Const x) = SOME x
2413 | findHeadConst (A $ B) = findHeadConst A
2414 | findHeadConst _ = NONE
2415fun findDef (Const (\<^const_name>‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2416 | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2417 | findDef (Abs (a,b,c)) = findDef c
2418 | findDef _ = NONE
2419val const_opt = (findDef trm)
2420val defs = case const_opt of SOME const => List.filter (fn thm => let
2421 val concl = Thm.concl_of thm
2422 val thmconst = (findDef concl)
2423 in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2424 | _ => []
2425in
2426resolve_tac ctxt defs 1
2427end
2428)) 1)))))›
2429‹Resolve AOT definitions›
2430
2431method AOT_subst_intro_helper = ((rule AOT_substI
2432 | AOT_defI
2433 | (simp only: AOT_subst_cond_𝗈_def AOT_subst_cond_fun_def; ((rule allI)+)?)))
2434
2435method AOT_subst for ψ::"'a::AOT_subst" and χ::"'a::AOT_subst" =
2436 (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2437 ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2438 ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2439 ; (AOT_subst_intro_helper+)?››)
2440
2441method AOT_subst_rev for χ::"'a::AOT_subst" and ψ::"'a::AOT_subst" =
2442 (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2443 ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2444 ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=χ and χ=ψ, THEN "≡E"(1)]
2445 ; (AOT_subst_intro_helper+)?››)
2446
2447method AOT_subst_manual for φ::"'a::AOT_subst ⇒ 𝗈" =
2448 (rule AOT_subst[where φ=φ, THEN "≡E"(2)]; (AOT_subst_intro_helper+)?)
2449
2450method AOT_subst_manual_rev for φ::"'a::AOT_subst ⇒ 𝗈" =
2451 (rule AOT_subst[where φ=φ, THEN "≡E"(1)]; (AOT_subst_intro_helper+)?)
2452
2453method AOT_subst_using uses subst =
2454 (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2455 match conclusion in "[v ⊨ «φ ψ»]" for φ v ⇒ ‹
2456 rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2457 ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2458
2459method AOT_subst_using_rev uses subst =
2460 (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2461 match conclusion in "[v ⊨ «φ χ»]" for φ v ⇒ ‹
2462 rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(1)]
2463 ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2464
2465AOT_theorem "rule-sub-remark:1[1]": assumes ‹❙⊢⇩□ A!x ≡ ¬◇E!x› and ‹¬A!x› shows ‹¬¬◇E!x›
2466 by (AOT_subst_rev "«A!x»" "«¬◇E!x»") (auto simp: assms)
2467
2468AOT_theorem "rule-sub-remark:1[2]": assumes ‹❙⊢⇩□ A!x ≡ ¬◇E!x› and ‹¬¬◇E!x› shows ‹¬A!x›
2469 by (AOT_subst "«A!x»" "«¬◇E!x»") (auto simp: assms)
2470
2471AOT_theorem "rule-sub-remark:2[1]":
2472 assumes ‹❙⊢⇩□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy› shows ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)›
2473 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2474
2475AOT_theorem "rule-sub-remark:2[2]":
2476 assumes ‹❙⊢⇩□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)› shows ‹p → [R]xy›
2477 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2478
2479AOT_theorem "rule-sub-remark:3[1]":
2480 assumes ‹for arbitrary x: ❙⊢⇩□ A!x ≡ ¬◇E!x›
2481 and ‹∃x A!x›
2482 shows ‹∃x ¬◇E!x›
2483 by (AOT_subst_rev "λκ. «A!κ»" "λκ. «¬◇E!κ»") (auto simp: assms)
2484
2485AOT_theorem "rule-sub-remark:3[2]":
2486 assumes ‹for arbitrary x: ❙⊢⇩□ A!x ≡ ¬◇E!x›
2487 and ‹∃x ¬◇E!x›
2488 shows ‹∃x A!x›
2489 by (AOT_subst "λκ. «A!κ»" "λκ. «¬◇E!κ»") (auto simp: assms)
2490
2491AOT_theorem "rule-sub-remark:4[1]":
2492 assumes ‹❙⊢⇩□ ¬¬[P]x ≡ [P]x› and ‹❙𝒜¬¬[P]x› shows ‹❙𝒜[P]x›
2493 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2494
2495AOT_theorem "rule-sub-remark:4[2]":
2496 assumes ‹❙⊢⇩□ ¬¬[P]x ≡ [P]x› and ‹❙𝒜[P]x› shows ‹❙𝒜¬¬[P]x›
2497 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2498
2499AOT_theorem "rule-sub-remark:5[1]":
2500 assumes ‹❙⊢⇩□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(φ → ψ)› shows ‹□(¬ψ → ¬φ)›
2501 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2502
2503AOT_theorem "rule-sub-remark:5[2]":
2504 assumes ‹❙⊢⇩□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(¬ψ → ¬φ)› shows ‹□(φ → ψ)›
2505 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2506
2507AOT_theorem "rule-sub-remark:6[1]":
2508 assumes ‹❙⊢⇩□ ψ ≡ χ› and ‹□(φ → ψ)› shows ‹□(φ → χ)›
2509 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2510
2511AOT_theorem "rule-sub-remark:6[2]":
2512 assumes ‹❙⊢⇩□ ψ ≡ χ› and ‹□(φ → χ)› shows ‹□(φ → ψ)›
2513 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2514
2515AOT_theorem "rule-sub-remark:7[1]":
2516 assumes ‹❙⊢⇩□ φ ≡ ¬¬φ› and ‹□(φ → φ)› shows ‹□(¬¬φ → φ)›
2517 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2518
2519AOT_theorem "rule-sub-remark:7[2]":
2520 assumes ‹❙⊢⇩□ φ ≡ ¬¬φ› and ‹□(¬¬φ → φ)› shows ‹□(φ → φ)›
2521 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2522
2523AOT_theorem "KBasic2:1": ‹□¬φ ≡ ¬◇φ›
2524 by (meson "conventions:5" "contraposition:2" "Hypothetical Syllogism" "df-rules-formulas[3]"
2525 "df-rules-formulas[4]" "≡I" "useful-tautologies:1")
2526
2527AOT_theorem "KBasic2:2": ‹◇(φ ∨ ψ) ≡ (◇φ ∨ ◇ψ)›
2528proof -
2529 AOT_have ‹◇(φ ∨ ψ) ≡ ◇¬(¬φ & ¬ψ)›
2530 by (simp add: "RE◇" "oth-class-taut:5:b")
2531 also AOT_have ‹… ≡ ¬□(¬φ & ¬ψ)›
2532 using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2533 also AOT_have ‹… ≡ ¬(□¬φ & □¬ψ)›
2534 using "KBasic:3" "≡E"(1) "oth-class-taut:4:b" by blast
2535 also AOT_have ‹… ≡ ¬(¬◇φ & ¬◇ψ)›
2536 apply (AOT_subst_rev "«□¬φ»" "«¬◇φ»")
2537 apply (simp add: "KBasic2:1")
2538 apply (AOT_subst_rev "«□¬ψ»" "«¬◇ψ»")
2539 by (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2540 also AOT_have ‹… ≡ ¬¬(◇φ ∨ ◇ψ)›
2541 using "≡E"(6) "oth-class-taut:3:b" "oth-class-taut:5:b" by blast
2542 also AOT_have ‹… ≡ ◇φ ∨ ◇ψ›
2543 by (simp add: "≡I" "useful-tautologies:1" "useful-tautologies:2")
2544 finally show ?thesis .
2545qed
2546
2547AOT_theorem "KBasic2:3": ‹◇(φ & ψ) → (◇φ & ◇ψ)›
2548 by (metis "RM◇" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "deduction-theorem" "modus-tollens:1" "reductio-aa:1")
2549
2550AOT_theorem "KBasic2:4": ‹◇(φ → ψ) ≡ (□φ → ◇ψ)›
2551proof -
2552 AOT_have ‹◇(φ → ψ) ≡ ◇(¬φ ∨ ψ)›
2553 by (AOT_subst "«φ → ψ»" "«¬φ ∨ ψ»")
2554 (auto simp: "oth-class-taut:1:c" "oth-class-taut:3:a")
2555 also AOT_have ‹... ≡ ◇¬φ ∨ ◇ψ›
2556 by (simp add: "KBasic2:2")
2557 also AOT_have ‹... ≡ ¬□φ ∨ ◇ψ›
2558 by (AOT_subst "«¬□φ»" "«◇¬φ»")
2559 (auto simp: "KBasic:11" "oth-class-taut:3:a")
2560 also AOT_have ‹... ≡ □φ → ◇ψ›
2561 using "≡E"(6) "oth-class-taut:1:c" "oth-class-taut:3:a" by blast
2562 finally show ?thesis .
2563qed
2564
2565AOT_theorem "KBasic2:5": ‹◇◇φ ≡ ¬□□¬φ›
2566 apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2567 apply (simp add: "conventions:5" "≡Df")
2568 apply (AOT_subst "«◇¬□¬φ»" "«¬□¬¬□¬φ»")
2569 apply (simp add: "conventions:5" "≡Df")
2570 apply (AOT_subst_rev "«□¬φ»" "«¬¬□¬φ»")
2571 apply (simp add: "oth-class-taut:3:b")
2572 by (simp add: "oth-class-taut:3:a")
2573
2574
2575AOT_theorem "KBasic2:6": ‹□(φ ∨ ψ) → (□φ ∨ ◇ψ)›
2576proof(rule "→I"; rule "raa-cor:1")
2577 AOT_assume ‹□(φ ∨ ψ)›
2578 AOT_hence ‹□(¬φ → ψ)›
2579 apply - apply (AOT_subst_rev "«φ ∨ ψ»" "«¬φ → ψ»")
2580 by (simp add: "conventions:2" "≡Df")
2581 AOT_hence 1: ‹◇¬φ → ◇ψ› using "KBasic:13" "vdash-properties:10" by blast
2582 AOT_assume ‹¬(□φ ∨ ◇ψ)›
2583 AOT_hence ‹¬□φ› and ‹¬◇ψ› using "&E" "≡E"(1) "oth-class-taut:5:d" by blast+
2584 AOT_thus ‹◇ψ & ¬◇ψ› using "&I"(1) 1[THEN "→E"] "KBasic:11" "≡E"(4) "raa-cor:3" by blast
2585qed
2586
2587AOT_theorem "KBasic2:7": ‹(□(φ ∨ ψ) & ◇¬φ) → ◇ψ›
2588proof(rule "→I"; frule "&E"(1); drule "&E"(2))
2589 AOT_assume ‹□(φ ∨ ψ)›
2590 AOT_hence 1: ‹□φ ∨ ◇ψ›
2591 using "KBasic2:6" "∨I"(2) "∨E"(1) by blast
2592 AOT_assume ‹◇¬φ›
2593 AOT_hence ‹¬□φ› using "KBasic:11" "≡E"(2) by blast
2594 AOT_thus ‹◇ψ› using 1 "∨E"(2) by blast
2595qed
2596
2597AOT_theorem "T-S5-fund:1": ‹φ → ◇φ›
2598 by (meson "≡⇩d⇩fI" "conventions:5" "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" "qml:2"[axiom_inst])
2599lemmas "T◇" = "T-S5-fund:1"
2600
2601AOT_theorem "T-S5-fund:2": ‹◇□φ → □φ›
2602proof(rule "→I")
2603 AOT_assume ‹◇□φ›
2604 AOT_hence ‹¬□◇¬φ›
2605 using "KBasic:14" "≡E"(4) "raa-cor:3" by blast
2606 moreover AOT_have ‹◇¬φ → □◇¬φ›
2607 by (fact "qml:3"[axiom_inst])
2608 ultimately AOT_have ‹¬◇¬φ›
2609 using "modus-tollens:1" by blast
2610 AOT_thus ‹□φ› using "KBasic:12" "≡E"(2) by blast
2611qed
2612lemmas "5◇" = "T-S5-fund:2"
2613
2614
2615AOT_theorem "Act-Sub:1": ‹❙𝒜φ ≡ ¬❙𝒜¬φ›
2616 by (AOT_subst "«❙𝒜¬φ»" "«¬❙𝒜φ»")
2617 (auto simp: "logic-actual-nec:1"[axiom_inst] "oth-class-taut:3:b")
2618
2619AOT_theorem "Act-Sub:2": ‹◇φ ≡ ❙𝒜◇φ›
2620 apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2621 apply (simp add: "conventions:5" "≡Df")
2622 by (metis "deduction-theorem" "≡I" "≡E"(1) "≡E"(2) "≡E"(3)
2623 "logic-actual-nec:1"[axiom_inst] "qml-act:2"[axiom_inst])
2624
2625AOT_theorem "Act-Sub:3": ‹❙𝒜φ → ◇φ›
2626 apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2627 apply (simp add: "conventions:5" "≡Df")
2628 by (metis "Act-Sub:1" "deduction-theorem" "≡E"(4) "nec-imp-act" "reductio-aa:2" "vdash-properties:6")
2629
2630
2631AOT_theorem "Act-Sub:4": ‹❙𝒜φ ≡ ◇❙𝒜φ›
2632proof (rule "≡I"; rule "→I")
2633 AOT_assume ‹❙𝒜φ›
2634 AOT_thus ‹◇❙𝒜φ› using "T◇" "vdash-properties:10" by blast
2635next
2636 AOT_assume ‹◇❙𝒜φ›
2637 AOT_hence ‹¬□¬❙𝒜φ›
2638 using "≡⇩d⇩fE" "conventions:5" by blast
2639 AOT_hence ‹¬□❙𝒜¬φ›
2640 apply - apply (AOT_subst "«❙𝒜¬φ»" "«¬❙𝒜φ»")
2641 by (simp add: "logic-actual-nec:1"[axiom_inst])
2642 AOT_thus ‹❙𝒜φ›
2643 using "Act-Basic:1" "Act-Basic:6" "∨E"(3) "≡E"(4) "reductio-aa:1" by blast
2644qed
2645
2646AOT_theorem "Act-Sub:5": ‹◇❙𝒜φ → ❙𝒜◇φ›
2647 by (metis "Act-Sub:2" "Act-Sub:3" "Act-Sub:4" "deduction-theorem" "≡E"(1) "≡E"(2) "vdash-properties:6")
2648
2649AOT_theorem "S5Basic:1": ‹◇φ ≡ □◇φ›
2650 by (simp add: "≡I" "qml:2" "qml:3" "vdash-properties:1[2]")
2651
2652AOT_theorem "S5Basic:2": ‹□φ ≡ ◇□φ›
2653 by (simp add: "T◇" "5◇" "≡I")
2654
2655AOT_theorem "S5Basic:3": ‹φ → □◇φ›
2656 using "T◇" "Hypothetical Syllogism" "qml:3" "vdash-properties:1[2]" by blast
2657lemmas "B" = "S5Basic:3"
2658
2659AOT_theorem "S5Basic:4": ‹◇□φ → φ›
2660 using "5◇" "Hypothetical Syllogism" "qml:2" "vdash-properties:1[2]" by blast
2661lemmas "B◇" = "S5Basic:4"
2662
2663AOT_theorem "S5Basic:5": ‹□φ → □□φ›
2664 using "RM:1" "B" "5◇" "Hypothetical Syllogism" by blast
2665lemmas "4" = "S5Basic:5"
2666
2667AOT_theorem "S5Basic:6": ‹□φ ≡ □□φ›
2668 by (simp add: "4" "≡I" "qml:2"[axiom_inst])
2669
2670AOT_theorem "S5Basic:7": ‹◇◇φ → ◇φ›
2671 apply (AOT_subst "«◇◇φ»" "«¬□¬◇φ»")
2672 apply (simp add: "conventions:5" "≡Df")
2673 apply (AOT_subst "«◇φ»" "«¬□¬φ»")
2674 apply (simp add: "conventions:5" "≡Df")
2675 apply (AOT_subst_rev "«□¬φ»" "«¬¬□¬φ»")
2676 apply (simp add: "oth-class-taut:3:b")
2677 apply (AOT_subst_rev "«□¬φ»" "«□□¬φ»")
2678 apply (simp add: "S5Basic:6")
2679 by (simp add: "if-p-then-p")
2680
2681lemmas "4◇" = "S5Basic:7"
2682
2683AOT_theorem "S5Basic:8": ‹◇◇φ ≡ ◇φ›
2684 by (simp add: "4◇" "T◇" "≡I")
2685
2686AOT_theorem "S5Basic:9": ‹□(φ ∨ □ψ) ≡ (□φ ∨ □ψ)›
2687 apply (rule "≡I"; rule "→I")
2688 using "KBasic2:6" "5◇" "∨I"(3) "if-p-then-p" "vdash-properties:10" apply blast
2689 by (meson "KBasic:15" "4" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "con-dis-taut:7"
2690 "intro-elim:1" "Commutativity of ∨")
2691
2692AOT_theorem "S5Basic:10": ‹□(φ ∨ ◇ψ) ≡ (□φ ∨ ◇ψ)›
2693
2694proof(rule "≡I"; rule "→I")
2695 AOT_assume ‹□(φ ∨ ◇ψ)›
2696 AOT_hence ‹□φ ∨ ◇◇ψ›
2697 by (meson "KBasic2:6" "∨I"(2) "∨E"(1))
2698 AOT_thus ‹□φ ∨ ◇ψ›
2699 by (meson "B◇" "4" "4◇" "T◇" "∨I"(3))
2700next
2701 AOT_assume ‹□φ ∨ ◇ψ›
2702 AOT_hence ‹□φ ∨ □◇ψ›
2703 by (meson "S5Basic:1" "B◇" "S5Basic:6" "T◇" "5◇" "∨I"(3) "intro-elim:1")
2704 AOT_thus ‹□(φ ∨ ◇ψ)›
2705 by (meson "KBasic:15" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
2706qed
2707
2708AOT_theorem "S5Basic:11": ‹◇(φ & ◇ψ) ≡ (◇φ & ◇ψ)›
2709proof -
2710 AOT_have ‹◇(φ & ◇ψ) ≡ ◇¬(¬φ ∨ ¬◇ψ)›
2711 by (AOT_subst "«φ & ◇ψ»" "«¬(¬φ ∨ ¬◇ψ)»")
2712 (auto simp: "oth-class-taut:5:a" "oth-class-taut:3:a")
2713 also AOT_have ‹… ≡ ◇¬(¬φ ∨ □¬ψ)›
2714 by (AOT_subst "«□¬ψ»" "«¬◇ψ»")
2715 (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2716 also AOT_have ‹… ≡ ¬□(¬φ ∨ □¬ψ)›
2717 using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2718 also AOT_have ‹… ≡ ¬(□¬φ ∨ □¬ψ)›
2719 using "S5Basic:9" "≡E"(1) "oth-class-taut:4:b" by blast
2720 also AOT_have ‹… ≡ ¬(¬◇φ ∨ ¬◇ψ)›
2721 apply (AOT_subst "«□¬φ»" "«¬◇φ»")
2722 apply (simp add: "KBasic2:1")
2723 apply (AOT_subst "«□¬ψ»" "«¬◇ψ»")
2724 by (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2725 also AOT_have ‹… ≡ ◇φ & ◇ψ›
2726 using "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:a" by blast
2727 finally show ?thesis .
2728qed
2729
2730AOT_theorem "S5Basic:12": ‹◇(φ & □ψ) ≡ (◇φ & □ψ)›
2731proof (rule "≡I"; rule "→I")
2732 AOT_assume ‹◇(φ & □ψ)›
2733 AOT_hence ‹◇φ & ◇□ψ›
2734 using "KBasic2:3" "vdash-properties:6" by blast
2735 AOT_thus ‹◇φ & □ψ›
2736 using "5◇" "&I" "&E"(1) "&E"(2) "vdash-properties:6" by blast
2737next
2738 AOT_assume ‹◇φ & □ψ›
2739 moreover AOT_have ‹(□□ψ & ◇φ) → ◇(φ & □ψ)›
2740 by (AOT_subst "«φ & □ψ»" "«□ψ & φ»")
2741 (auto simp: "Commutativity of &" "KBasic:16")
2742 ultimately AOT_show ‹◇(φ & □ψ)›
2743 by (metis "4" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2744qed
2745
2746
2747AOT_theorem "S5Basic:13": ‹□(φ → □ψ) ≡ □(◇φ → ψ)›
2748proof (rule "≡I")
2749 AOT_modally_strict {
2750 AOT_have ‹□(φ → □ψ) → (◇φ → ψ)›
2751 by (meson "KBasic:13" "B◇" "Hypothetical Syllogism" "deduction-theorem")
2752 }
2753 AOT_hence ‹□□(φ → □ψ) → □(◇φ → ψ)›
2754 by (rule RM)
2755 AOT_thus ‹□(φ → □ψ) → □(◇φ → ψ)›
2756 using "4" "Hypothetical Syllogism" by blast
2757next
2758 AOT_modally_strict {
2759 AOT_have ‹□(◇φ → ψ) → (φ → □ψ)›
2760 by (meson "B" "Hypothetical Syllogism" "deduction-theorem" "qml:1" "vdash-properties:1[2]")
2761 }
2762 AOT_hence ‹□□(◇φ → ψ) → □(φ → □ψ)›
2763 by (rule RM)
2764 AOT_thus ‹□(◇φ → ψ) → □(φ → □ψ)›
2765 using "4" "Hypothetical Syllogism" by blast
2766qed
2767
2768AOT_theorem "derived-S5-rules:1":
2769 assumes ‹Γ ❙⊢⇩□ ◇φ → ψ› shows ‹□Γ ❙⊢⇩□ φ → □ψ›
2770proof -
2771 AOT_have ‹□Γ ❙⊢⇩□ □◇φ → □ψ›
2772 using assms by (rule "RM:1[prem]")
2773 AOT_thus ‹□Γ ❙⊢⇩□ φ → □ψ›
2774 using "B" "Hypothetical Syllogism" by blast
2775qed
2776
2777AOT_theorem "derived-S5-rules:2":
2778 assumes ‹Γ ❙⊢⇩□ φ → □ψ› shows ‹□Γ ❙⊢⇩□ ◇φ → ψ›
2779proof -
2780 AOT_have ‹□Γ ❙⊢⇩□ ◇φ → ◇□ψ›
2781 using assms by (rule "RM:2[prem]")
2782 AOT_thus ‹□Γ ❙⊢⇩□ ◇φ → ψ›
2783 using "B◇" "Hypothetical Syllogism" by blast
2784qed
2785
2786AOT_theorem "BFs:1": ‹∀α □φ{α} → □∀α φ{α}›
2787proof -
2788 AOT_modally_strict {
2789 AOT_modally_strict {
2790 AOT_have ‹∀α □φ{α} → □φ{α}› for α by (fact AOT)
2791 }
2792 AOT_hence ‹◇∀α □φ{α} → ◇□φ{α}› for α by (rule "RM◇")
2793 AOT_hence ‹◇∀α □φ{α} → ∀α φ{α}›
2794 using "B◇" "∀I" "→E" "→I" by metis
2795 }
2796 thus ?thesis using "derived-S5-rules:1" by blast
2797qed
2798lemmas "BF" = "BFs:1"
2799
2800AOT_theorem "BFs:2": ‹□∀α φ{α} → ∀α □φ{α}›
2801proof -
2802 AOT_have ‹□∀α φ{α} → □φ{α}› for α using RM "cqt-orig:3" by metis
2803 thus ?thesis using "cqt-orig:2"[THEN "→E"] "∀I" by metis
2804qed
2805lemmas "CBF" = "BFs:2"
2806
2807AOT_theorem "BFs:3": ‹◇∃α φ{α} → ∃α ◇φ{α}›
2808proof(rule "→I")
2809 AOT_modally_strict {
2810 AOT_have ‹□∀α ¬φ{α} ≡ ∀α □¬φ{α}›
2811 using BF CBF "≡I" by blast
2812 } note θ = this
2813
2814 AOT_assume ‹◇∃α φ{α}›
2815 AOT_hence ‹¬□¬(∃α φ{α})›
2816 using "≡⇩d⇩fE" "conventions:5" by blast
2817 AOT_hence ‹¬□∀α ¬φ{α}›
2818 apply - apply (AOT_subst "«∀α ¬φ{α}»" "«¬(∃α φ{α})»")
2819 using "≡⇩d⇩fI" "conventions:3" "conventions:4" "&I" "contraposition:2" "cqt-further:4"
2820 "df-rules-formulas[1]" "vdash-properties:1[2]" by blast
2821 AOT_hence ‹¬∀α □¬φ{α}›
2822 apply - apply (AOT_subst_using_rev subst: θ)
2823 using θ by blast
2824 AOT_hence ‹¬∀α ¬¬□¬φ{α}›
2825 apply - apply (AOT_subst_rev "λ τ. «□¬φ{τ}»" "λ τ. «¬¬□¬φ{τ}»")
2826 by (simp add: "oth-class-taut:3:b")
2827 AOT_hence 0: ‹∃α ¬□¬φ{α}›
2828 by (rule "conventions:4"[THEN "≡⇩d⇩fI"])
2829 AOT_show ‹∃α ◇φ{α}›
2830 apply (AOT_subst "λ τ . «◇φ{τ}»" "λ τ . «¬□¬φ{τ}»")
2831 apply (simp add: "conventions:5" "≡Df")
2832 using 0 by blast
2833qed
2834lemmas "BF◇" = "BFs:3"
2835
2836AOT_theorem "BFs:4": ‹∃α ◇φ{α} → ◇∃α φ{α}›
2837proof(rule "→I")
2838 AOT_assume ‹∃α ◇φ{α}›
2839 AOT_hence ‹¬∀α ¬◇φ{α}›
2840 using "conventions:4"[THEN "≡⇩d⇩fE"] by blast
2841 AOT_hence ‹¬∀α □¬φ{α}›
2842 apply - apply (AOT_subst "λ τ . «□¬φ{τ}»" "λ τ . «¬◇φ{τ}»")
2843 by (simp add: "KBasic2:1")
2844 moreover AOT_have ‹∀α □¬φ{α} ≡ □∀α ¬φ{α}›
2845 using "≡I" "BF" "CBF" by metis
2846 ultimately AOT_have 1: ‹¬□∀α ¬φ{α}›
2847 using "≡E"(3) by blast
2848 AOT_show ‹◇∃α φ{α}›
2849 apply (rule "conventions:5"[THEN "≡⇩d⇩fI"])
2850 apply (AOT_subst "«∃α φ{α}»" "«¬∀α ¬φ{α}»")
2851 apply (simp add: "conventions:4" "≡Df")
2852 apply (AOT_subst "«¬¬∀α ¬φ{α}»" "«∀α ¬φ{α}»")
2853 by (auto simp: 1 "≡I" "useful-tautologies:1" "useful-tautologies:2")
2854qed
2855lemmas "CBF◇" = "BFs:4"
2856
2857AOT_theorem "sign-S5-thm:1": ‹∃α □φ{α} → □∃α φ{α}›
2858proof(rule "→I")
2859 AOT_assume ‹∃α □φ{α}›
2860 then AOT_obtain α where ‹□φ{α}› using "∃E" by metis
2861 moreover AOT_have ‹□α↓›
2862 by (simp add: "ex:1:a" "rule-ui:2[const_var]" RN)
2863 moreover AOT_have ‹□φ{τ}, □τ↓ ❙⊢⇩□ □∃α φ{α}› for τ
2864 proof -
2865 AOT_have ‹φ{τ}, τ↓ ❙⊢⇩□ ∃α φ{α}› using "existential:1" by blast
2866 AOT_thus ‹□φ{τ}, □τ↓ ❙⊢⇩□ □∃α φ{α}›
2867 using "RN[prem]"[where Γ="{φ τ, «τ↓»}", simplified] by blast
2868 qed
2869 ultimately AOT_show ‹□∃α φ{α}› by blast
2870qed
2871lemmas Buridan = "sign-S5-thm:1"
2872
2873AOT_theorem "sign-S5-thm:2": ‹◇∀α φ{α} → ∀α ◇φ{α}›
2874proof -
2875 AOT_have ‹∀α (◇∀α φ{α} → ◇φ{α})›
2876 by (simp add: "RM◇" "cqt-orig:3" "∀I")
2877 AOT_thus ‹◇∀α φ{α} → ∀α ◇φ{α}›
2878 using "∀E"(4) "∀I" "→E" "→I" by metis
2879qed
2880lemmas "Buridan◇" = "sign-S5-thm:2"
2881
2882AOT_theorem "sign-S5-thm:3": ‹◇∃α (φ{α} & ψ{α}) → ◇(∃α φ{α} & ∃α ψ{α})›
2883 apply (rule "RM:2")
2884 by (metis (no_types, lifting) "instantiation" "&I" "&E"(1)
2885 "&E"(2) "deduction-theorem" "existential:2[const_var]")
2886
2887AOT_theorem "sign-S5-thm:4": ‹◇∃α (φ{α} & ψ{α}) → ◇∃α φ{α}›
2888 apply (rule "RM:2")
2889 by (meson "instantiation" "&E"(1) "deduction-theorem" "existential:2[const_var]")
2890
2891AOT_theorem "sign-S5-thm:5": ‹(□∀α (φ{α} → ψ{α}) & □∀α (ψ{α} → χ{α})) → □∀α (φ{α} → χ{α})›
2892proof -
2893 {
2894 fix φ' ψ' χ'
2895 AOT_assume ‹❙⊢⇩□ φ' & ψ' → χ'›
2896 AOT_hence ‹□φ' & □ψ' → □χ'›
2897 using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2898 using "&E" "&I" "→E" "→I" by metis
2899 } note R = this
2900 show ?thesis by (rule R; fact AOT)
2901qed
2902
2903AOT_theorem "sign-S5-thm:6": ‹(□∀α (φ{α} ≡ ψ{α}) & □∀α(ψ{α} ≡ χ{α})) → □∀α(φ{α} ≡ χ{α})›
2904proof -
2905 {
2906 fix φ' ψ' χ'
2907 AOT_assume ‹❙⊢⇩□ φ' & ψ' → χ'›
2908 AOT_hence ‹□φ' & □ψ' → □χ'›
2909 using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2910 using "&E" "&I" "→E" "→I" by metis
2911 } note R = this
2912 show ?thesis by (rule R; fact AOT)
2913qed
2914
2915AOT_theorem "exist-nec2:1": ‹◇τ↓ → τ↓›
2916 using "B◇" "RM◇" "Hypothetical Syllogism" "exist-nec" by blast
2917
2918AOT_theorem "exists-nec2:2": ‹◇τ↓ ≡ □τ↓›
2919 by (meson "Act-Sub:3" "Hypothetical Syllogism" "exist-nec" "exist-nec2:1" "≡I" "nec-imp-act")
2920
2921AOT_theorem "exists-nec2:3": ‹¬τ↓ → □¬τ↓›
2922 using "KBasic2:1" "deduction-theorem" "exist-nec2:1" "≡E"(2) "modus-tollens:1" by blast
2923
2924AOT_theorem "exists-nec2:4": ‹◇¬τ↓ ≡ □¬τ↓›
2925 by (metis "Act-Sub:3" "KBasic:12" "deduction-theorem" "exist-nec" "exists-nec2:3" "≡I" "≡E"(4) "nec-imp-act" "reductio-aa:1")
2926
2927AOT_theorem "id-nec2:1": ‹◇α = β → α = β›
2928 using "B◇" "RM◇" "Hypothetical Syllogism" "id-nec:1" by blast
2929
2930AOT_theorem "id-nec2:2": ‹α ≠ β → □α ≠ β›
2931 apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
2932 using "KBasic2:1" "deduction-theorem" "id-nec2:1" "≡E"(2) "modus-tollens:1" by blast
2933
2934AOT_theorem "id-nec2:3": ‹◇α ≠ β → α ≠ β›
2935 apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
2936 by (metis "KBasic:11" "deduction-theorem" "id-nec:2" "≡E"(3) "reductio-aa:2" "vdash-properties:6")
2937
2938AOT_theorem "id-nec2:4": ‹◇α = β → □α = β›
2939 using "Hypothetical Syllogism" "id-nec2:1" "id-nec:1" by blast
2940
2941AOT_theorem "id-nec2:5": ‹◇α ≠ β → □α ≠ β›
2942 using "id-nec2:3" "id-nec2:2" "→I" "→E" by metis
2943
2944AOT_theorem "sc-eq-box-box:1": ‹□(φ → □φ) ≡ (◇φ → □φ)›
2945 apply (rule "≡I"; rule "→I")
2946 using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:10" apply blast
2947 by (metis "KBasic2:1" "KBasic:1" "KBasic:2" "S5Basic:13" "≡E"(2) "raa-cor:5" "vdash-properties:6")
2948
2949AOT_theorem "sc-eq-box-box:2": ‹(□(φ → □φ) ∨ (◇φ → □φ)) → (◇φ ≡ □φ)›
2950 by (metis "Act-Sub:3" "KBasic:13" "5◇" "∨E"(2) "deduction-theorem" "≡I" "nec-imp-act" "raa-cor:2" "vdash-properties:10")
2951
2952AOT_theorem "sc-eq-box-box:3": ‹□(φ → □φ) → (¬□φ ≡ □¬φ)›
2953proof (rule "→I"; rule "≡I"; rule "→I")
2954 AOT_assume ‹□(φ → □φ)›
2955 AOT_hence ‹◇φ → □φ› using "sc-eq-box-box:1" "≡E" by blast
2956 moreover AOT_assume ‹¬□φ›
2957 ultimately AOT_have ‹¬◇φ›
2958 using "modus-tollens:1" by blast
2959 AOT_thus ‹□¬φ›
2960 using "KBasic2:1" "≡E"(2) by blast
2961next
2962 AOT_assume ‹□(φ → □φ)›
2963 moreover AOT_assume ‹□¬φ›
2964 ultimately AOT_show ‹¬□φ›
2965 using "modus-tollens:1" "qml:2" "vdash-properties:10" "vdash-properties:1[2]" by blast
2966qed
2967
2968AOT_theorem "sc-eq-box-box:4": ‹(□(φ → □φ) & □(ψ → □ψ)) → ((□φ ≡ □ψ) → □(φ ≡ ψ))›
2969proof(rule "→I"; rule "→I")
2970 AOT_assume θ: ‹□(φ → □φ) & □(ψ → □ψ)›
2971 AOT_assume ξ: ‹□φ ≡ □ψ›
2972 AOT_hence ‹(□φ & □ψ) ∨ (¬□φ & ¬□ψ)›
2973 using "≡E"(4) "oth-class-taut:4:g" "raa-cor:3" by blast
2974 moreover {
2975 AOT_assume ‹□φ & □ψ›
2976 AOT_hence ‹□(φ ≡ ψ)›
2977 using "KBasic:3" "KBasic:8" "≡E"(2) "vdash-properties:10" by blast
2978 }
2979 moreover {
2980 AOT_assume ‹¬□φ & ¬□ψ›
2981 moreover AOT_have ‹¬□φ ≡ □¬φ› and ‹¬□ψ ≡ □¬ψ›
2982 using θ "Conjunction Simplification"(1) "Conjunction Simplification"(2) "sc-eq-box-box:3" "vdash-properties:10" by metis+
2983 ultimately AOT_have ‹□¬φ & □¬ψ›
2984 by (metis "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "≡E"(4) "modus-tollens:1" "raa-cor:3")
2985 AOT_hence ‹□(φ ≡ ψ)›
2986 using "KBasic:3" "KBasic:9" "≡E"(2) "vdash-properties:10" by blast
2987 }
2988 ultimately AOT_show ‹□(φ ≡ ψ)›
2989 using "∨E"(2) "reductio-aa:1" by blast
2990qed
2991
2992AOT_theorem "sc-eq-box-box:5": ‹(□(φ → □φ) & □(ψ → □ψ)) → ((φ ≡ ψ) → □(φ ≡ ψ))›
2993proof (rule "→I"; rule "→I")
2994 AOT_assume A: ‹(□(φ → □φ) & □(ψ → □ψ))›
2995 AOT_hence ‹φ → □φ› and ‹ψ → □ψ›
2996 using "&E" "qml:2"[axiom_inst] "→E" by blast+
2997 moreover AOT_assume ‹φ ≡ ψ›
2998 ultimately AOT_have ‹□φ ≡ □ψ›
2999 using "→E" "qml:2"[axiom_inst] "≡E" "≡I" by meson
3000 moreover AOT_have ‹(□φ ≡ □ψ) → □(φ ≡ ψ)›
3001 using A "sc-eq-box-box:4" "→E" by blast
3002 ultimately AOT_show ‹□(φ ≡ ψ)› using "→E" by blast
3003qed
3004
3005AOT_theorem "sc-eq-box-box:6": ‹□(φ → □φ) → ((φ → □ψ) → □(φ → ψ))›
3006proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3007 AOT_assume ‹¬□(φ → ψ)›
3008 AOT_hence 1: ‹◇¬(φ → ψ)› by (metis "KBasic:11" "≡E"(1))
3009 AOT_have ‹◇(φ & ¬ψ)›
3010 apply (AOT_subst ‹«φ & ¬ψ»› ‹«¬(φ → ψ)»›)
3011 apply (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3012 by (fact 1)
3013 AOT_hence ‹◇φ› and 2: ‹◇¬ψ› using "KBasic2:3"[THEN "→E"] "&E" by blast+
3014 moreover AOT_assume ‹□(φ → □φ)›
3015 ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3016 AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3017 moreover AOT_assume ‹φ → □ψ›
3018 ultimately AOT_have ‹□ψ› using "→E" by blast
3019 moreover AOT_have ‹¬□ψ› using 2 "KBasic:12" "¬¬I" "intro-elim:3:d" by blast
3020 ultimately AOT_show ‹□ψ & ¬□ψ› using "&I" by blast
3021qed
3022
3023AOT_theorem "sc-eq-box-box:7": ‹□(φ → □φ) → ((φ → ❙𝒜ψ) → ❙𝒜(φ → ψ))›
3024proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3025 AOT_assume ‹¬❙𝒜(φ → ψ)›
3026 AOT_hence 1: ‹❙𝒜¬(φ → ψ)› by (metis "Act-Basic:1" "∨E"(2))
3027 AOT_have ‹❙𝒜(φ & ¬ψ)›
3028 apply (AOT_subst ‹«φ & ¬ψ»› ‹«¬(φ → ψ)»›)
3029 apply (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3030 by (fact 1)
3031 AOT_hence ‹❙𝒜φ› and 2: ‹❙𝒜¬ψ› using "Act-Basic:2"[THEN "≡E"(1)] "&E" by blast+
3032 AOT_hence ‹◇φ› by (metis "Act-Sub:3" "→E")
3033 moreover AOT_assume ‹□(φ → □φ)›
3034 ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3035 AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3036 moreover AOT_assume ‹φ → ❙𝒜ψ›
3037 ultimately AOT_have ‹❙𝒜ψ› using "→E" by blast
3038 moreover AOT_have ‹¬❙𝒜ψ› using 2 by (meson "Act-Sub:1" "≡E"(4) "raa-cor:3")
3039 ultimately AOT_show ‹❙𝒜ψ & ¬❙𝒜ψ› using "&I" by blast
3040qed
3041
3042AOT_theorem "sc-eq-fur:1": ‹◇❙𝒜φ ≡ □❙𝒜φ›
3043 using "Act-Basic:6" "Act-Sub:4" "≡E"(6) by blast
3044
3045AOT_theorem "sc-eq-fur:2": ‹□(φ → □φ) → (❙𝒜φ ≡ φ)›
3046 by (metis "B◇" "Act-Sub:3" "KBasic:13" "T◇" "Hypothetical Syllogism" "deduction-theorem" "≡I" "nec-imp-act")
3047
3048AOT_theorem "sc-eq-fur:3": ‹□∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ❙ιx φ{x}↓)›
3049proof (rule "→I"; rule "→I")
3050 AOT_assume ‹□∀x (φ{x} → □φ{x})›
3051 AOT_hence A: ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3052 AOT_assume ‹∃!x φ{x}›
3053 then AOT_obtain a where a_def: ‹φ{a} & ∀y (φ{y} → y = a)›
3054 using "∃E"[rotated 1, OF "uniqueness:1"[THEN "≡⇩d⇩fE"]] by blast
3055 moreover AOT_have ‹□φ{a}› using calculation A "∀E"(2) "qml:2"[axiom_inst] "→E" "&E"(1) by blast
3056 AOT_hence ‹❙𝒜φ{a}› using "nec-imp-act" "vdash-properties:6" by blast
3057 moreover AOT_have ‹∀y (❙𝒜φ{y} → y = a)›
3058 proof (rule "∀I"; rule "→I")
3059 fix b
3060 AOT_assume ‹❙𝒜φ{b}›
3061 AOT_hence ‹◇φ{b}›
3062 using "Act-Sub:3" "vdash-properties:6" by blast
3063 moreover {
3064 AOT_have ‹□(φ{b} → □φ{b})›
3065 using A "∀E"(2) by blast
3066 AOT_hence ‹◇φ{b} → □φ{b}›
3067 using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:6" by blast
3068 }
3069 ultimately AOT_have ‹□φ{b}› using "→E" by blast
3070 AOT_hence ‹φ{b}› using "qml:2"[axiom_inst] "→E" by blast
3071 AOT_thus ‹b = a›
3072 using a_def[THEN "&E"(2)] "∀E"(2) "→E" by blast
3073 qed
3074 ultimately AOT_have ‹❙𝒜φ{a} & ∀y (❙𝒜φ{y} → y = a)›
3075 using "&I" by blast
3076 AOT_hence ‹∃x (❙𝒜φ{x} & ∀y (❙𝒜φ{y} → y = x))› using "∃I" by fast
3077 AOT_hence ‹∃!x ❙𝒜φ{x}› using "uniqueness:1"[THEN "≡⇩d⇩fI"] by fast
3078 AOT_thus ‹❙ιx φ{x}↓›
3079 using "actual-desc:1"[THEN "≡E"(2)] by blast
3080qed
3081
3082AOT_theorem "sc-eq-fur:4": ‹□∀x (φ{x} → □φ{x}) → (x = ❙ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x)))›
3083proof (rule "→I")
3084 AOT_assume ‹□∀x (φ{x} → □φ{x})›
3085 AOT_hence ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3086 AOT_hence A: ‹❙𝒜φ{α} ≡ φ{α}› for α using "sc-eq-fur:2" "∀E" "→E" by fast
3087 AOT_show ‹x = ❙ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x))›
3088 proof (rule "≡I"; rule "→I")
3089 AOT_assume ‹x = ❙ιx φ{x}›
3090 AOT_hence B: ‹❙𝒜φ{x} & ∀z (❙𝒜φ{z} → z = x)›
3091 using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
3092 AOT_show ‹φ{x} & ∀z (φ{z} → z = x)›
3093 proof (rule "&I"; (rule "∀I"; rule "→I")?)
3094 AOT_show ‹φ{x}› using A B[THEN "&E"(1)] "≡E"(1) by blast
3095 next
3096 AOT_show ‹z = x› if ‹φ{z}› for z
3097 using that B[THEN "&E"(2)] "∀E"(2) "→E" A[THEN "≡E"(2)] by blast
3098 qed
3099 next
3100 AOT_assume B: ‹φ{x} & ∀z (φ{z} → z = x)›
3101 AOT_have ‹❙𝒜φ{x} & ∀z (❙𝒜φ{z} → z = x)›
3102 proof(rule "&I"; (rule "∀I"; rule "→I")?)
3103 AOT_show ‹❙𝒜φ{x}› using B[THEN "&E"(1)] A[THEN "≡E"(2)] by blast
3104 next
3105 AOT_show ‹b = x› if ‹❙𝒜φ{b}› for b
3106 using that A[THEN "≡E"(1)] B[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] by blast
3107 qed
3108 AOT_thus ‹x = ❙ιx φ{x}›
3109 using "nec-hintikka-scheme"[THEN "≡E"(2)] by blast
3110 qed
3111qed
3112
3113AOT_theorem "id-act:1": ‹α = β ≡ ❙𝒜α = β›
3114 by (meson "Act-Sub:3" "Hypothetical Syllogism" "id-nec2:1" "id-nec:2" "≡I" "nec-imp-act")
3115
3116AOT_theorem "id-act:2": ‹α ≠ β ≡ ❙𝒜α ≠ β›
3117proof (AOT_subst "«α ≠ β»" "«¬(α = β)»")
3118 AOT_modally_strict {
3119 AOT_show ‹α ≠ β ≡ ¬(α = β)›
3120 by (simp add: "=-infix" "≡Df")
3121 }
3122next
3123 AOT_show ‹¬(α = β) ≡ ❙𝒜¬(α = β)›
3124 proof (safe intro!: "≡I" "→I")
3125 AOT_assume ‹¬α = β›
3126 AOT_hence ‹¬❙𝒜α = β› using "id-act:1" "≡E"(3) by blast
3127 AOT_thus ‹❙𝒜¬α = β›
3128 using "¬¬E" "Act-Sub:1" "≡E"(3) by blast
3129 next
3130 AOT_assume ‹❙𝒜¬α = β›
3131 AOT_hence ‹¬❙𝒜α = β›
3132 using "¬¬I" "Act-Sub:1" "≡E"(4) by blast
3133 AOT_thus ‹¬α = β›
3134 using "id-act:1" "≡E"(4) by blast
3135 qed
3136qed
3137
3138AOT_theorem "A-Exists:1": ‹❙𝒜∃!α φ{α} ≡ ∃!α ❙𝒜φ{α}›
3139proof -
3140 AOT_have ‹❙𝒜∃!α φ{α} ≡ ❙𝒜∃α∀β (φ{β} ≡ β = α)›
3141 by (AOT_subst_using subst: "uniqueness:2")
3142 (simp add: "oth-class-taut:3:a")
3143 also AOT_have ‹… ≡ ∃α ❙𝒜∀β (φ{β} ≡ β = α)›
3144 by (simp add: "Act-Basic:10")
3145 also AOT_have ‹… ≡ ∃α∀β ❙𝒜(φ{β} ≡ β = α)›
3146 by (AOT_subst "λ τ . «❙𝒜∀β (φ{β} ≡ β = τ)»" "λ τ . «∀β ❙𝒜(φ{β} ≡ β = τ)»")
3147 (auto simp: "logic-actual-nec:3" "vdash-properties:1[2]" "oth-class-taut:3:a")
3148 also AOT_have ‹… ≡ ∃α∀β (❙𝒜φ{β} ≡ ❙𝒜β = α)›
3149 by (AOT_subst_rev "λ τ τ' . «❙𝒜(φ{τ'} ≡ τ' = τ)»" "λ τ τ'. «❙𝒜φ{τ'} ≡ ❙𝒜τ' = τ»")
3150 (auto simp: "Act-Basic:5" "cqt-further:7")
3151 also AOT_have ‹… ≡ ∃α∀β (❙𝒜φ{β} ≡ β = α)›
3152 apply (AOT_subst "λ τ τ' :: 'a . «❙𝒜τ' = τ»" "λ τ τ'. «τ' = τ»")
3153 apply (meson "id-act:1" "≡E"(6) "oth-class-taut:3:a")
3154 by (simp add: "cqt-further:7")
3155 also AOT_have ‹... ≡ ∃!α ❙𝒜φ{α}›
3156 using "uniqueness:2" "Commutativity of ≡"[THEN "≡E"(1)] by fast
3157 finally show ?thesis .
3158qed
3159
3160AOT_theorem "A-Exists:2": ‹❙ιx φ{x}↓ ≡ ❙𝒜∃!x φ{x}›
3161 by (AOT_subst_using subst: "A-Exists:1")
3162 (simp add: "actual-desc:1")
3163
3164AOT_theorem "id-act-desc:1": ‹❙ιx (x = y)↓›
3165proof(rule "existence:1"[THEN "≡⇩d⇩fI"]; rule "∃I")
3166 AOT_show ‹[λx E!x → E!x]❙ιx (x = y)›
3167 proof (rule "russell-axiom[exe,1].nec-russell-axiom"[THEN "≡E"(2)]; rule "∃I"; (rule "&I")+)
3168 AOT_show ‹❙𝒜y = y› by (simp add: "RA[2]" "id-eq:1")
3169 next
3170 AOT_show ‹∀z (❙𝒜z = y → z = y)›
3171 apply (rule "∀I")
3172 using "id-act:1"[THEN "≡E"(2)] "→I" by blast
3173 next
3174 AOT_show ‹[λx E!x → E!x]y›
3175 proof (rule "lambda-predicates:2"[axiom_inst, THEN "→E", THEN "≡E"(2)])
3176 AOT_show ‹[λx E!x → E!x]↓›
3177 by "cqt:2[lambda]"
3178 next
3179 AOT_show ‹E!y → E!y›
3180 by (simp add: "if-p-then-p")
3181 qed
3182 qed
3183next
3184 AOT_show ‹[λx E!x → E!x]↓›
3185 by "cqt:2[lambda]"
3186qed
3187
3188AOT_theorem "id-act-desc:2": ‹y = ❙ιx (x = y)›
3189 by (rule descriptions[axiom_inst, THEN "≡E"(2)]; rule "∀I"; rule "id-act:1"[symmetric])
3190
3191AOT_theorem "pre-en-eq:1[1]": ‹x⇩1[F] → □x⇩1[F]›
3192 by (simp add: encoding "vdash-properties:1[2]")
3193
3194AOT_theorem "pre-en-eq:1[2]": ‹x⇩1x⇩2[F] → □x⇩1x⇩2[F]›
3195proof (rule "→I")
3196 AOT_assume ‹x⇩1x⇩2[F]›
3197 AOT_hence ‹x⇩1[λy [F]yx⇩2]› and ‹x⇩2[λy [F]x⇩1y]›
3198 using "nary-encoding[2]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3199 moreover AOT_have ‹[λy [F]yx⇩2]↓› by "cqt:2[lambda]"
3200 moreover AOT_have ‹[λy [F]x⇩1y]↓› by "cqt:2[lambda]"
3201 ultimately AOT_have ‹□x⇩1[λy [F]yx⇩2]› and ‹□x⇩2[λy [F]x⇩1y]›
3202 using encoding[axiom_inst, unvarify F] "→E" "&I" by blast+
3203 note A = this
3204 AOT_hence ‹□(x⇩1[λy [F]yx⇩2] & x⇩2[λy [F]x⇩1y])›
3205 using "KBasic:3"[THEN "≡E"(2)] "&I" by blast
3206 AOT_thus ‹□x⇩1x⇩2[F]›
3207 by (rule "nary-encoding[2]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3208qed
3209
3210AOT_theorem "pre-en-eq:1[3]": ‹x⇩1x⇩2x⇩3[F] → □x⇩1x⇩2x⇩3[F]›
3211proof (rule "→I")
3212 AOT_assume ‹x⇩1x⇩2x⇩3[F]›
3213 AOT_hence ‹x⇩1[λy [F]yx⇩2x⇩3]› and ‹x⇩2[λy [F]x⇩1yx⇩3]› and ‹x⇩3[λy [F]x⇩1x⇩2y]›
3214 using "nary-encoding[3]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3215 moreover AOT_have ‹[λy [F]yx⇩2x⇩3]↓› by "cqt:2[lambda]"
3216 moreover AOT_have ‹[λy [F]x⇩1yx⇩3]↓› by "cqt:2[lambda]"
3217 moreover AOT_have ‹[λy [F]x⇩1x⇩2y]↓› by "cqt:2[lambda]"
3218 ultimately AOT_have ‹□x⇩1[λy [F]yx⇩2x⇩3]› and ‹□x⇩2[λy [F]x⇩1yx⇩3]› and ‹□x⇩3[λy [F]x⇩1x⇩2y]›
3219 using encoding[axiom_inst, unvarify F] "→E" by blast+
3220 note A = this
3221 AOT_have B: ‹□(x⇩1[λy [F]yx⇩2x⇩3] & x⇩2[λy [F]x⇩1yx⇩3] & x⇩3[λy [F]x⇩1x⇩2y])›
3222 by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3223 AOT_thus ‹□x⇩1x⇩2x⇩3[F]›
3224 by (rule "nary-encoding[3]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3225qed
3226
3227AOT_theorem "pre-en-eq:1[4]": ‹x⇩1x⇩2x⇩3x⇩4[F] → □x⇩1x⇩2x⇩3x⇩4[F]›
3228proof (rule "→I")
3229 AOT_assume ‹x⇩1x⇩2x⇩3x⇩4[F]›
3230 AOT_hence ‹x⇩1[λy [F]yx⇩2x⇩3x⇩4]› and ‹x⇩2[λy [F]x⇩1yx⇩3x⇩4]› and ‹x⇩3[λy [F]x⇩1x⇩2yx⇩4]› and ‹x⇩4[λy [F]x⇩1x⇩2x⇩3y]›
3231 using "nary-encoding[4]"[axiom_inst, THEN "≡E"(1)] "&E" by metis+
3232 moreover AOT_have ‹[λy [F]yx⇩2x⇩3x⇩4]↓› by "cqt:2[lambda]"
3233 moreover AOT_have ‹[λy [F]x⇩1yx⇩3x⇩4]↓› by "cqt:2[lambda]"
3234 moreover AOT_have ‹[λy [F]x⇩1x⇩2yx⇩4]↓› by "cqt:2[lambda]"
3235 moreover AOT_have ‹[λy [F]x⇩1x⇩2x⇩3y]↓› by "cqt:2[lambda]"
3236 ultimately AOT_have ‹□x⇩1[λy [F]yx⇩2x⇩3x⇩4]› and ‹□x⇩2[λy [F]x⇩1yx⇩3x⇩4]› and ‹□x⇩3[λy [F]x⇩1x⇩2yx⇩4]› and ‹□x⇩4[λy [F]x⇩1x⇩2x⇩3y]›
3237 using "→E" encoding[axiom_inst, unvarify F] by blast+
3238 note A = this
3239 AOT_have B: ‹□(x⇩1[λy [F]yx⇩2x⇩3x⇩4] & x⇩2[λy [F]x⇩1yx⇩3x⇩4] & x⇩3[λy [F]x⇩1x⇩2yx⇩4] & x⇩4[λy [F]x⇩1x⇩2x⇩3y])›
3240 by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3241 AOT_thus ‹□x⇩1x⇩2x⇩3x⇩4[F]›
3242 by (rule "nary-encoding[4]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3243qed
3244
3245AOT_theorem "pre-en-eq:2[1]": ‹¬x⇩1[F] → □¬x⇩1[F]›
3246proof (rule "→I"; rule "raa-cor:1")
3247 AOT_assume ‹¬□¬x⇩1[F]›
3248 AOT_hence ‹◇x⇩1[F]›
3249 by (rule "conventions:5"[THEN "≡⇩d⇩fI"])
3250 AOT_hence ‹x⇩1[F]›
3251 by(rule "S5Basic:13"[THEN "≡E"(1), OF "pre-en-eq:1[1]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3252 moreover AOT_assume ‹¬x⇩1[F]›
3253 ultimately AOT_show ‹x⇩1[F] & ¬x⇩1[F]› by (rule "&I")
3254qed
3255AOT_theorem "pre-en-eq:2[2]": ‹¬x⇩1x⇩2[F] → □¬x⇩1x⇩2[F]›
3256proof (rule "→I"; rule "raa-cor:1")
3257 AOT_assume ‹¬□¬x⇩1x⇩2[F]›
3258 AOT_hence ‹◇x⇩1x⇩2[F]›
3259 by (rule "conventions:5"[THEN "≡⇩d⇩fI"])
3260 AOT_hence ‹x⇩1x⇩2[F]›
3261 by(rule "S5Basic:13"[THEN "≡E"(1), OF "pre-en-eq:1[2]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3262 moreover AOT_assume ‹¬x⇩1x⇩2[F]›
3263 ultimately AOT_show ‹x⇩1x⇩2[F] & ¬x⇩1x⇩2[F]› by (rule "&I")
3264qed
3265
3266AOT_theorem "pre-en-eq:2[3]": ‹¬x⇩1x⇩2x⇩3[F] → □¬x⇩1x⇩2x⇩3[F]›
3267proof (rule "→I"; rule "raa-cor:1")
3268 AOT_assume ‹¬□¬x⇩1x⇩2x⇩3[F]›
3269 AOT_hence ‹◇x⇩1x⇩2x⇩3[F]›
3270 by (rule "conventions:5"[THEN "≡⇩d⇩fI"])
3271 AOT_hence ‹x⇩1x⇩2x⇩3[F]›
3272 by(rule "S5Basic:13"[THEN "≡E"(1), OF "pre-en-eq:1[3]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3273 moreover AOT_assume ‹¬x⇩1x⇩2x⇩3[F]›
3274 ultimately AOT_show ‹x⇩1x⇩2x⇩3[F] & ¬x⇩1x⇩2x⇩3[F]› by (rule "&I")
3275qed
3276
3277AOT_theorem "pre-en-eq:2[4]": ‹¬x⇩1x⇩2x⇩3x⇩4[F] → □¬x⇩1x⇩2x⇩3x⇩4[F]›
3278proof (rule "→I"; rule "raa-cor:1")
3279 AOT_assume ‹¬□¬x⇩1x⇩2x⇩3x⇩4[F]›
3280 AOT_hence ‹◇x⇩1x⇩2x⇩3x⇩4[F]›
3281 by (rule "conventions:5"[THEN "≡⇩d⇩fI"])
3282 AOT_hence ‹x⇩1x⇩2x⇩3x⇩4[F]›
3283 by(rule "S5Basic:13"[THEN "≡E"(1), OF "pre-en-eq:1[4]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3284 moreover AOT_assume ‹¬x⇩1x⇩2x⇩3x⇩4[F]›
3285 ultimately AOT_show ‹x⇩1x⇩2x⇩3x⇩4[F] & ¬x⇩1x⇩2x⇩3x⇩4[F]› by (rule "&I")
3286qed
3287
3288AOT_theorem "en-eq:1[1]": ‹◇x⇩1[F] ≡ □x⇩1[F]›
3289 using "pre-en-eq:1[1]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3290AOT_theorem "en-eq:1[2]": ‹◇x⇩1x⇩2[F] ≡ □x⇩1x⇩2[F]›
3291 using "pre-en-eq:1[2]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3292AOT_theorem "en-eq:1[3]": ‹◇x⇩1x⇩2x⇩3[F] ≡ □x⇩1x⇩2x⇩3[F]›
3293 using "pre-en-eq:1[3]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3294AOT_theorem "en-eq:1[4]": ‹◇x⇩1x⇩2x⇩3x⇩4[F] ≡ □x⇩1x⇩2x⇩3x⇩4[F]›
3295 using "pre-en-eq:1[4]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3296
3297AOT_theorem "en-eq:2[1]": ‹x⇩1[F] ≡ □x⇩1[F]›
3298 by (simp add: "≡I" "pre-en-eq:1[1]" "qml:2"[axiom_inst])
3299AOT_theorem "en-eq:2[2]": ‹x⇩1x⇩2[F] ≡ □x⇩1x⇩2[F]›
3300 by (simp add: "≡I" "pre-en-eq:1[2]" "qml:2"[axiom_inst])
3301AOT_theorem "en-eq:2[3]": ‹x⇩1x⇩2x⇩3[F] ≡ □x⇩1x⇩2x⇩3[F]›
3302 by (simp add: "≡I" "pre-en-eq:1[3]" "qml:2"[axiom_inst])
3303AOT_theorem "en-eq:2[4]": ‹x⇩1x⇩2x⇩3x⇩4[F] ≡ □x⇩1x⇩2x⇩3x⇩4[F]›
3304 by (simp add: "≡I" "pre-en-eq:1[4]" "qml:2"[axiom_inst])
3305
3306AOT_theorem "en-eq:3[1]": ‹◇x⇩1[F] ≡ x⇩1[F]›
3307 using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[1]"] "≡I" by blast
3308AOT_theorem "en-eq:3[2]": ‹◇x⇩1x⇩2[F] ≡ x⇩1x⇩2[F]›
3309 using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[2]"] "≡I" by blast
3310AOT_theorem "en-eq:3[3]": ‹◇x⇩1x⇩2x⇩3[F] ≡ x⇩1x⇩2x⇩3[F]›
3311 using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[3]"] "≡I" by blast
3312AOT_theorem "en-eq:3[4]": ‹◇x⇩1x⇩2x⇩3x⇩4[F] ≡ x⇩1x⇩2x⇩3x⇩4[F]›
3313 using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[4]"] "≡I" by blast
3314
3315AOT_theorem "en-eq:4[1]": ‹(x⇩1[F] ≡ y⇩1[G]) ≡ (□x⇩1[F] ≡ □y⇩1[G])›
3316 apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3317 using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[1]" by blast+
3318AOT_theorem "en-eq:4[2]": ‹(x⇩1x⇩2[F] ≡ y⇩1y⇩2[G]) ≡ (□x⇩1x⇩2[F] ≡ □y⇩1y⇩2[G])›
3319 apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3320 using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[2]" by blast+
3321AOT_theorem "en-eq:4[3]": ‹(x⇩1x⇩2x⇩3[F] ≡ y⇩1y⇩2y⇩3[G]) ≡ (□x⇩1x⇩2x⇩3[F] ≡ □y⇩1y⇩2y⇩3[G])›
3322 apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3323 using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[3]" by blast+
3324AOT_theorem "en-eq:4[4]": ‹(x⇩1x⇩2x⇩3x⇩4[F] ≡ y⇩1y⇩2y⇩3y⇩4[G]) ≡ (□x⇩1x⇩2x⇩3x⇩4[F] ≡ □y⇩1y⇩2y⇩3y⇩4[G])›
3325 apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3326 using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[4]" by blast+
3327
3328AOT_theorem "en-eq:5[1]": ‹□(x⇩1[F] ≡ y⇩1[G]) ≡ (□x⇩1[F] ≡ □y⇩1[G])›
3329 apply (rule "≡I"; rule "→I")
3330 using "en-eq:4[1]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3331 using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3332 "&I"[OF "pre-en-eq:1[1]"[THEN RN], OF "pre-en-eq:1[1]"[THEN RN]] by blast
3333AOT_theorem "en-eq:5[2]": ‹□(x⇩1x⇩2[F] ≡ y⇩1y⇩2[G]) ≡ (□x⇩1x⇩2[F] ≡ □y⇩1y⇩2[G])›
3334 apply (rule "≡I"; rule "→I")
3335 using "en-eq:4[2]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3336 using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3337 "&I"[OF "pre-en-eq:1[2]"[THEN RN], OF "pre-en-eq:1[2]"[THEN RN]] by blast
3338AOT_theorem "en-eq:5[3]": ‹□(x⇩1x⇩2x⇩3[F] ≡ y⇩1y⇩2y⇩3[G]) ≡ (□x⇩1x⇩2x⇩3[F] ≡ □y⇩1y⇩2y⇩3[G])›
3339 apply (rule "≡I"; rule "→I")
3340 using "en-eq:4[3]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3341 using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3342 "&I"[OF "pre-en-eq:1[3]"[THEN RN], OF "pre-en-eq:1[3]"[THEN RN]] by blast
3343AOT_theorem "en-eq:5[4]": ‹□(x⇩1x⇩2x⇩3x⇩4[F] ≡ y⇩1y⇩2y⇩3y⇩4[G]) ≡ (□x⇩1x⇩2x⇩3x⇩4[F] ≡ □y⇩1y⇩2y⇩3y⇩4[G])›
3344 apply (rule "≡I"; rule "→I")
3345 using "en-eq:4[4]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3346 using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3347 "&I"[OF "pre-en-eq:1[4]"[THEN RN], OF "pre-en-eq:1[4]"[THEN RN]] by blast
3348
3349AOT_theorem "en-eq:6[1]": ‹(x⇩1[F] ≡ y⇩1[G]) ≡ □(x⇩1[F] ≡ y⇩1[G])›
3350 using "en-eq:5[1]"[symmetric] "en-eq:4[1]" "≡E"(5) by fast
3351AOT_theorem "en-eq:6[2]": ‹(x⇩1x⇩2[F] ≡ y⇩1y⇩2[G]) ≡ □(x⇩1x⇩2[F] ≡ y⇩1y⇩2[G])›
3352 using "en-eq:5[2]"[symmetric] "en-eq:4[2]" "≡E"(5) by fast
3353AOT_theorem "en-eq:6[3]": ‹(x⇩1x⇩2x⇩3[F] ≡ y⇩1y⇩2y⇩3[G]) ≡ □(x⇩1x⇩2x⇩3[F] ≡ y⇩1y⇩2y⇩3[G])›
3354 using "en-eq:5[3]"[symmetric] "en-eq:4[3]" "≡E"(5) by fast
3355AOT_theorem "en-eq:6[4]": ‹(x⇩1x⇩2x⇩3x⇩4[F] ≡ y⇩1y⇩2y⇩3y⇩4[G]) ≡ □(x⇩1x⇩2x⇩3x⇩4[F] ≡ y⇩1y⇩2y⇩3y⇩4[G])›
3356 using "en-eq:5[4]"[symmetric] "en-eq:4[4]" "≡E"(5) by fast
3357
3358AOT_theorem "en-eq:7[1]": ‹¬x⇩1[F] ≡ □¬x⇩1[F]›
3359 using "pre-en-eq:2[1]" "qml:2"[axiom_inst] "≡I" by blast
3360AOT_theorem "en-eq:7[2]": ‹¬x⇩1x⇩2[F] ≡ □¬x⇩1x⇩2[F]›
3361 using "pre-en-eq:2[2]" "qml:2"[axiom_inst] "≡I" by blast
3362AOT_theorem "en-eq:7[3]": ‹¬x⇩1x⇩2x⇩3[F] ≡ □¬x⇩1x⇩2x⇩3[F]›
3363 using "pre-en-eq:2[3]" "qml:2"[axiom_inst] "≡I" by blast
3364AOT_theorem "en-eq:7[4]": ‹¬x⇩1x⇩2x⇩3x⇩4[F] ≡ □¬x⇩1x⇩2x⇩3x⇩4[F]›
3365 using "pre-en-eq:2[4]" "qml:2"[axiom_inst] "≡I" by blast
3366
3367AOT_theorem "en-eq:8[1]": ‹◇¬x⇩1[F] ≡ ¬x⇩1[F]›
3368 using "en-eq:2[1]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3369AOT_theorem "en-eq:8[2]": ‹◇¬x⇩1x⇩2[F] ≡ ¬x⇩1x⇩2[F]›
3370 using "en-eq:2[2]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3371AOT_theorem "en-eq:8[3]": ‹◇¬x⇩1x⇩2x⇩3[F] ≡ ¬x⇩1x⇩2x⇩3[F]›
3372 using "en-eq:2[3]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3373AOT_theorem "en-eq:8[4]": ‹◇¬x⇩1x⇩2x⇩3x⇩4[F] ≡ ¬x⇩1x⇩2x⇩3x⇩4[F]›
3374 using "en-eq:2[4]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3375
3376AOT_theorem "en-eq:9[1]": ‹◇¬x⇩1[F] ≡ □¬x⇩1[F]›
3377 using "en-eq:7[1]" "en-eq:8[1]" "≡E"(5) by blast
3378AOT_theorem "en-eq:9[2]": ‹◇¬x⇩1x⇩2[F] ≡ □¬x⇩1x⇩2[F]›
3379 using "en-eq:7[2]" "en-eq:8[2]" "≡E"(5) by blast
3380AOT_theorem "en-eq:9[3]": ‹◇¬x⇩1x⇩2x⇩3[F] ≡ □¬x⇩1x⇩2x⇩3[F]›
3381 using "en-eq:7[3]" "en-eq:8[3]" "≡E"(5) by blast
3382AOT_theorem "en-eq:9[4]": ‹◇¬x⇩1x⇩2x⇩3x⇩4[F] ≡ □¬x⇩1x⇩2x⇩3x⇩4[F]›
3383 using "en-eq:7[4]" "en-eq:8[4]" "≡E"(5) by blast
3384
3385AOT_theorem "en-eq:10[1]": ‹❙𝒜x⇩1[F] ≡ x⇩1[F]›
3386 by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[1]" "pre-en-eq:1[1]")
3387AOT_theorem "en-eq:10[2]": ‹❙𝒜x⇩1x⇩2[F] ≡ x⇩1x⇩2[F]›
3388 by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[2]" "pre-en-eq:1[2]")
3389AOT_theorem "en-eq:10[3]": ‹❙𝒜x⇩1x⇩2x⇩3[F] ≡ x⇩1x⇩2x⇩3[F]›
3390 by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[3]" "pre-en-eq:1[3]")
3391AOT_theorem "en-eq:10[4]": ‹❙𝒜x⇩1x⇩2x⇩3x⇩4[F] ≡ x⇩1x⇩2x⇩3x⇩4[F]›
3392 by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[4]" "pre-en-eq:1[4]")
3393
3394AOT_theorem "oa-facts:1": ‹O!x → □O!x›
3395proof(rule "→I")
3396 AOT_modally_strict {
3397 AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
3398 by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3399 } note θ = this
3400 AOT_assume ‹O!x›
3401 AOT_hence ‹[λx ◇E!x]x›
3402 by (rule "=⇩d⇩fE"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3403 AOT_hence ‹◇E!x› using θ[THEN "≡E"(1)] by blast
3404 AOT_hence 0: ‹□◇E!x› using "qml:3"[axiom_inst, THEN "→E"] by blast
3405 AOT_have ‹□[λx ◇E!x]x›
3406 by (AOT_subst_using subst: θ) (simp add: 0)
3407 AOT_thus ‹□O!x›
3408 by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3409qed
3410
3411AOT_theorem "oa-facts:2": ‹A!x → □A!x›
3412proof(rule "→I")
3413 AOT_modally_strict {
3414 AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
3415 by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3416 } note θ = this
3417 AOT_assume ‹A!x›
3418 AOT_hence ‹[λx ¬◇E!x]x›
3419 by (rule "=⇩d⇩fE"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3420 AOT_hence ‹¬◇E!x› using θ[THEN "≡E"(1)] by blast
3421 AOT_hence ‹□¬E!x› using "KBasic2:1"[THEN "≡E"(2)] by blast
3422 AOT_hence 0: ‹□□¬E!x› using "4"[THEN "→E"] by blast
3423 AOT_have 1: ‹□¬◇E!x›
3424 apply (AOT_subst "«¬◇E!x»" "«□¬E!x»")
3425 using "KBasic2:1"[symmetric] apply blast
3426 using 0 by blast
3427 AOT_have ‹□[λx ¬◇E!x]x›
3428 by (AOT_subst_using subst: θ) (simp add: 1)
3429 AOT_thus ‹□A!x›
3430 by (rule "=⇩d⇩fI"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3431qed
3432
3433AOT_theorem "oa-facts:3": ‹◇O!x → O!x›
3434 using "oa-facts:1" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3435AOT_theorem "oa-facts:4": ‹◇A!x → A!x›
3436 using "oa-facts:2" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3437
3438AOT_theorem "oa-facts:5": ‹◇O!x ≡ □O!x›
3439 by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3440
3441AOT_theorem "oa-facts:6": ‹◇A!x ≡ □A!x›
3442 by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3443
3444AOT_theorem "oa-facts:7": ‹O!x ≡ ❙𝒜O!x›
3445 by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3446
3447AOT_theorem "oa-facts:8": ‹A!x ≡ ❙𝒜A!x›
3448 by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3449
3450AOT_theorem "beta-C-meta": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n, ν⇩1...ν⇩n}]↓ → ([λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n, ν⇩1...ν⇩n}]ν⇩1...ν⇩n ≡ φ{ν⇩1...ν⇩n, ν⇩1...ν⇩n})›
3451 using "lambda-predicates:2"[axiom_inst] by blast
3452
3453AOT_theorem "beta-C-cor:1": ‹(∀ν⇩1...∀ν⇩n([λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n, ν⇩1...ν⇩n}]↓)) → ∀ν⇩1...∀ν⇩n ([λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n, ν⇩1...ν⇩n}]ν⇩1...ν⇩n ≡ φ{ν⇩1...ν⇩n, ν⇩1...ν⇩n})›
3454 apply (rule "cqt-basic:14"[where 'a='a, THEN "→E"])
3455 using "beta-C-meta" "∀I" by fast
3456
3457AOT_theorem "beta-C-cor:2": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓ → ∀ν⇩1...∀ν⇩n ([λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]ν⇩1...ν⇩n ≡ φ{ν⇩1...ν⇩n})›
3458 apply (rule "→I"; rule "∀I")
3459 using "beta-C-meta"[THEN "→E"] by fast
3460
3461
3462theorem "beta-C-cor:3": assumes ‹⋀ν⇩1ν⇩n. AOT_instance_of_cqt_2 (φ (AOT_term_of_var ν⇩1ν⇩n))›
3463 shows ‹[v ⊨ ∀ν⇩1...∀ν⇩n ([λμ⇩1...μ⇩n φ{ν⇩1...ν⇩n,μ⇩1...μ⇩n}]ν⇩1...ν⇩n ≡ φ{ν⇩1...ν⇩n,ν⇩1...ν⇩n})]›
3464 using "cqt:2[lambda]"[axiom_inst, OF assms] "beta-C-cor:1"[THEN "→E"] "∀I" by fast
3465
3466AOT_theorem "betaC:1:a": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n ❙⊢⇩□ φ{κ⇩1...κ⇩n}›
3467proof -
3468 AOT_modally_strict {
3469 AOT_assume ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3470 moreover AOT_have ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓› and ‹κ⇩1...κ⇩n↓›
3471 using calculation "cqt:5:a"[axiom_inst, THEN "→E"] "&E" by blast+
3472 ultimately AOT_show ‹φ{κ⇩1...κ⇩n}›
3473 using "beta-C-cor:2"[THEN "→E", THEN "∀E"(1), THEN "≡E"(1)] by blast
3474 }
3475qed
3476
3477AOT_theorem "betaC:1:b": ‹¬φ{κ⇩1...κ⇩n} ❙⊢⇩□ ¬[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3478 using "betaC:1:a" "raa-cor:3" by blast
3479
3480lemmas "β→C" = "betaC:1:a" "betaC:1:b"
3481
3482AOT_theorem "betaC:2:a": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓, κ⇩1...κ⇩n↓, φ{κ⇩1...κ⇩n} ❙⊢⇩□ [λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3483proof -
3484 AOT_modally_strict {
3485 AOT_assume 1: ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓› and 2: ‹κ⇩1...κ⇩n↓› and 3: ‹φ{κ⇩1...κ⇩n}›
3486 AOT_hence ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3487 using "beta-C-cor:2"[THEN "→E", OF 1, THEN "∀E"(1), THEN "≡E"(2)] by blast
3488 }
3489 AOT_thus ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓, κ⇩1...κ⇩n↓, φ{κ⇩1...κ⇩n} ❙⊢⇩□ [λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3490 by blast
3491qed
3492
3493AOT_theorem "betaC:2:b": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓, κ⇩1...κ⇩n↓, ¬[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n ❙⊢⇩□ ¬φ{κ⇩1...κ⇩n}›
3494 using "betaC:2:a" "raa-cor:3" by blast
3495
3496lemmas "β←C" = "betaC:2:a" "betaC:2:b"
3497
3498AOT_theorem "eta-conversion-lemma1:1": ‹Π↓ → [λx⇩1...x⇩n [Π]x⇩1...x⇩n] = Π›
3499 using "lambda-predicates:3"[axiom_inst] "∀I" "∀E"(1) "→I" by fast
3500
3501AOT_theorem "eta-conversion-lemma1:2": ‹Π↓ → [λν⇩1...ν⇩n [Π]ν⇩1...ν⇩n] = Π›
3502 using "eta-conversion-lemma1:1".
3503
3504
3505
3506text‹Note: not explicitly part of PLM.›
3507AOT_theorem id_sym: assumes ‹τ = τ'› shows ‹τ' = τ›
3508 using "rule=E"[where φ="λ τ' . «τ' = τ»", rotated 1, OF assms]
3509 "=I"(1)[OF "t=t-proper:1"[THEN "→E", OF assms]] by auto
3510declare id_sym[sym]
3511
3512text‹Note: not explicitly part of PLM.›
3513AOT_theorem id_trans: assumes ‹τ = τ'› and ‹τ' = τ''› shows ‹τ = τ''›
3514 using "rule=E" assms by blast
3515declare id_trans[trans]
3516
3517method "ηC" for Π :: ‹<'a::{AOT_Term_id_2,AOT_κs}>› = (match conclusion in "[v ⊨ τ{Π} = τ'{Π}]" for v τ τ' ⇒ ‹
3518rule "rule=E"[rotated 1, OF "eta-conversion-lemma1:2"[THEN "→E", of v "«[Π]»", symmetric]]
3519›)
3520
3526
3527
3528AOT_theorem "sub-des-lam:1": ‹[λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}]↓ & ❙ιx φ{x} = ❙ιx ψ{x} → [λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}] = [λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx ψ{x}}]›
3529proof(rule "→I")
3530 AOT_assume A: ‹[λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}]↓ & ❙ιx φ{x} = ❙ιx ψ{x}›
3531 AOT_show ‹[λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}] = [λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx ψ{x}}]›
3532 using "rule=E"[where φ="λ τ . «[λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}] = [λz⇩1...z⇩n χ{z⇩1...z⇩n, τ}]»",
3533 OF "=I"(1)[OF A[THEN "&E"(1)]], OF A[THEN "&E"(2)]]
3534 by blast
3535qed
3536
3537AOT_theorem "sub-des-lam:2": ‹❙ιx φ{x} = ❙ιx ψ{x} → χ{❙ιx φ{x}} = χ{❙ιx ψ{x}}› for χ :: ‹κ ⇒ 𝗈›
3538 using "rule=E"[where φ="λ τ . «χ{❙ιx φ{x}} = χ{τ}»", OF "=I"(1)[OF "log-prop-prop:2"]] "→I" by blast
3539
3540AOT_theorem "prop-equiv": ‹F = G ≡ ∀x (x[F] ≡ x[G])›
3541proof(rule "≡I"; rule "→I")
3542 AOT_assume ‹F = G›
3543 AOT_thus ‹∀x (x[F] ≡ x[G])›
3544 by (rule "rule=E"[rotated]) (fact "oth-class-taut:3:a"[THEN GEN])
3545next
3546 AOT_assume ‹∀x (x[F] ≡ x[G])›
3547 AOT_hence ‹x[F] ≡ x[G]› for x using "∀E" by blast
3548 AOT_hence ‹□(x[F] ≡ x[G])› for x using "en-eq:6[1]"[THEN "≡E"(1)] by blast
3549 AOT_hence ‹∀x □(x[F] ≡ x[G])› by (rule GEN)
3550 AOT_hence ‹□∀x (x[F] ≡ x[G])› using BF[THEN "→E"] by fast
3551 AOT_thus "F = G" using "p-identity-thm2:1"[THEN "≡E"(2)] by blast
3552qed
3553
3554AOT_theorem "relations:1":
3555 assumes ‹INSTANCE_OF_CQT_2(φ)›
3556 shows ‹∃F □∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ φ{x⇩1...x⇩n})›
3557 apply (rule "∃I"(1)[where τ="«[λx⇩1...x⇩n φ{x⇩1...x⇩n}]»"])
3558 using "cqt:2[lambda]"[OF assms, axiom_inst] "beta-C-cor:2"[THEN "→E", THEN RN] by blast+
3559
3560AOT_theorem "relations:2":
3561 assumes ‹INSTANCE_OF_CQT_2(φ)›
3562 shows ‹∃F □∀x ([F]x ≡ φ{x})›
3563 using "relations:1" assms by blast
3564
3565AOT_theorem "block-paradox:1": ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3566proof(rule RAA(2))
3567 let ?φ="λ τ. «∃G (τ[G] & ¬[G]τ)»"
3568 AOT_assume A: ‹[λx «?φ x»]↓›
3569 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ F = [λx «?φ x»]))›
3570 using "A-objects"[axiom_inst] by fast
3571 then AOT_obtain a where ξ: ‹A!a & ∀F (a[F] ≡ F = [λx «?φ x»])›
3572 using "∃E"[rotated] by blast
3573 AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3574 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3575 AOT_assume B: ‹[λx «?φ x»]a›
3576 AOT_hence ‹∃G (a[G] & ¬[G]a)› using "β→C" A by blast
3577 then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3578 moreover AOT_have ‹P = [λx «?φ x»]›
3579 using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3580 ultimately AOT_have ‹¬[λx «?φ x»]a›
3581 using "rule=E" "&E"(2) by fast
3582 AOT_thus ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B RAA by blast
3583 next
3584 AOT_assume B: ‹¬[λx «?φ x»]a›
3585 AOT_hence ‹¬∃G (a[G] & ¬[G]a)› using "β←C" "cqt:2[const_var]"[of a, axiom_inst] A by blast
3586 AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3587 AOT_have ‹∀G (a[G] → [G]a)›
3588 by (AOT_subst "λ Π . «a[Π] → [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3589 (auto simp: "oth-class-taut:1:a" C)
3590 AOT_hence ‹a[λx «?φ x»] → [λx «?φ x»]a› using "∀E" A by blast
3591 moreover AOT_have ‹a[λx «?φ x»]› using ξ[THEN "&E"(2), THEN "∀E"(1), OF A, THEN "≡E"(2)]
3592 using "=I"(1)[OF A] by blast
3593 ultimately AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B "→E" RAA by blast
3594 qed
3595qed(simp)
3596
3597AOT_theorem "block-paradox:2": ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3598proof(rule RAA(2))
3599 AOT_assume ‹∃F ∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
3600 then AOT_obtain F where F_prop: ‹∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))› using "∃E"[rotated] by blast
3601 AOT_have ‹∃x (A!x & ∀G (x[G] ≡ G = F))›
3602 using "A-objects"[axiom_inst] by fast
3603 then AOT_obtain a where ξ: ‹A!a & ∀G (a[G] ≡ G = F)›
3604 using "∃E"[rotated] by blast
3605 AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3606 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3607 AOT_assume B: ‹[F]a›
3608 AOT_hence ‹∃G (a[G] & ¬[G]a)› using F_prop[THEN "∀E"(2), THEN "≡E"(1)] by blast
3609 then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3610 moreover AOT_have ‹P = F›
3611 using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3612 ultimately AOT_have ‹¬[F]a›
3613 using "rule=E" "&E"(2) by fast
3614 AOT_thus ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B RAA by blast
3615 next
3616 AOT_assume B: ‹¬[F]a›
3617 AOT_hence ‹¬∃G (a[G] & ¬[G]a)›
3618 using "oth-class-taut:4:b"[THEN "≡E"(1), OF F_prop[THEN "∀E"(2)[of _ _ a]], THEN "≡E"(1)] by simp
3619 AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3620 AOT_have ‹∀G (a[G] → [G]a)›
3621 by (AOT_subst "λ Π . «a[Π] → [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3622 (auto simp: "oth-class-taut:1:a" C)
3623 AOT_hence ‹a[F] → [F]a› using "∀E" by blast
3624 moreover AOT_have ‹a[F]› using ξ[THEN "&E"(2), THEN "∀E"(2), of F, THEN "≡E"(2)]
3625 using "=I"(2) by blast
3626 ultimately AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B "→E" RAA by blast
3627 qed
3628qed(simp)
3629
3630AOT_theorem "block-paradox:3": ‹¬∀y [λz z = y]↓›
3631proof(rule RAA(2))
3632 AOT_assume θ: ‹∀y [λz z = y]↓›
3633 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃y(F = [λz z = y] & ¬y[F])))›
3634 using "A-objects"[axiom_inst] by force
3635 then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y (F = [λz z = y] & ¬y[F]))›
3636 using "∃E"[rotated] by blast
3637 AOT_have ζ: ‹a[λz z = a] ≡ ∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3638 using θ[THEN "∀E"(2)] a_prop[THEN "&E"(2), THEN "∀E"(1)] by blast
3639 AOT_show ‹¬∀y [λz z = y]↓›
3640 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3641 AOT_assume A: ‹a[λz z = a]›
3642 AOT_hence ‹∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3643 using ζ[THEN "≡E"(1)] by blast
3644 then AOT_obtain b where b_prop: ‹[λz z = a] = [λz z = b] & ¬b[λz z = a]›
3645 using "∃E"[rotated] by blast
3646 moreover AOT_have ‹a = a› by (rule "=I")
3647 moreover AOT_have ‹[λz z = a]↓› using θ "∀E" by blast
3648 moreover AOT_have ‹a↓› using "cqt:2[const_var]"[axiom_inst] .
3649 ultimately AOT_have ‹[λz z = a]a› using "β←C" by blast
3650 AOT_hence ‹[λz z = b]a› using "rule=E" b_prop[THEN "&E"(1)] by fast
3651 AOT_hence ‹a = b› using "β→C" by blast
3652 AOT_hence ‹b[λz z = a]› using A "rule=E" by fast
3653 AOT_thus ‹¬∀y [λz z = y]↓› using b_prop[THEN "&E"(2)] RAA by blast
3654 next
3655 AOT_assume A: ‹¬a[λz z = a]›
3656 AOT_hence ‹¬∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3657 using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3658 AOT_hence ‹∀y ¬([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3659 using "cqt-further:4"[THEN "→E"] by blast
3660 AOT_hence ‹¬([λz z = a] = [λz z = a] & ¬a[λz z = a])›
3661 using "∀E" by blast
3662 AOT_hence ‹[λz z = a] = [λz z = a] → a[λz z = a]›
3663 by (metis "&I" "deduction-theorem" "raa-cor:4")
3664 AOT_hence ‹a[λz z = a]› using "=I"(1) θ[THEN "∀E"(2)] "→E" by blast
3665 AOT_thus ‹¬∀y [λz z = y]↓› using A RAA by blast
3666 qed
3667qed(simp)
3668
3669AOT_theorem "block-paradox:4": ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3670proof(rule RAA(2))
3671 AOT_assume θ: ‹∀y ∃F ∀x([F]x ≡ x = y)›
3672 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F])))›
3673 using "A-objects"[axiom_inst] by force
3674 then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F]))›
3675 using "∃E"[rotated] by blast
3676 AOT_obtain F where F_prop: ‹∀x ([F]x ≡ x = a)› using θ[THEN "∀E"(2)] "∃E"[rotated] by blast
3677 AOT_have ζ: ‹a[F] ≡ ∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3678 using a_prop[THEN "&E"(2), THEN "∀E"(2)] by blast
3679 AOT_show ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3680 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3681 AOT_assume A: ‹a[F]›
3682 AOT_hence ‹∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3683 using ζ[THEN "≡E"(1)] by blast
3684 then AOT_obtain b where b_prop: ‹∀y ([F]y ≡ y = b) & ¬b[F]›
3685 using "∃E"[rotated] by blast
3686 moreover AOT_have ‹[F]a› using F_prop[THEN "∀E"(2), THEN "≡E"(2)] "=I"(2) by blast
3687 ultimately AOT_have ‹a = b› using "∀E"(2) "≡E"(1) "&E" by fast
3688 AOT_hence ‹a = b› using "β→C" by blast
3689 AOT_hence ‹b[F]› using A "rule=E" by fast
3690 AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using b_prop[THEN "&E"(2)] RAA by blast
3691 next
3692 AOT_assume A: ‹¬a[F]›
3693 AOT_hence ‹¬∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3694 using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3695 AOT_hence ‹∀z ¬(∀y ([F]y ≡ y = z) & ¬z[F])›
3696 using "cqt-further:4"[THEN "→E"] by blast
3697 AOT_hence ‹¬(∀y ([F]y ≡ y = a) & ¬a[F])›
3698 using "∀E" by blast
3699 AOT_hence ‹∀y ([F]y ≡ y = a) → a[F]›
3700 by (metis "&I" "deduction-theorem" "raa-cor:4")
3701 AOT_hence ‹a[F]› using F_prop "→E" by blast
3702 AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using A RAA by blast
3703 qed
3704qed(simp)
3705
3706AOT_theorem "block-paradox:5": ‹¬∃F∀x∀y([F]xy ≡ y = x)›
3707proof(rule "raa-cor:2")
3708 AOT_assume ‹∃F∀x∀y([F]xy ≡ y = x)›
3709 then AOT_obtain F where F_prop: ‹∀x∀y([F]xy ≡ y = x)› using "∃E"[rotated] by blast
3710 {
3711 fix x
3712 AOT_have 1: ‹∀y([F]xy ≡ y = x)› using F_prop "∀E" by blast
3713 AOT_have 2: ‹[λz [F]xz]↓› by "cqt:2[lambda]"
3714 moreover AOT_have ‹∀y([λz [F]xz]y ≡ y = x)›
3715 proof(rule "∀I")
3716 fix y
3717 AOT_have ‹[λz [F]xz]y ≡ [F]xy›
3718 using "beta-C-meta"[THEN "→E"] 2 by fast
3719 also AOT_have ‹... ≡ y = x› using 1 "∀E"
3720 by fast
3721 finally AOT_show ‹[λz [F]xz]y ≡ y = x›.
3722 qed
3723 ultimately AOT_have ‹∃F∀y([F]y ≡ y = x)›
3724 using "∃I" by fast
3725 }
3726 AOT_hence ‹∀x∃F∀y([F]y ≡ y = x)›
3727 by (rule GEN)
3728 AOT_thus ‹∀x∃F∀y([F]y ≡ y = x) & ¬∀x∃F∀y([F]y ≡ y = x)›
3729 using "&I" "block-paradox:4" by blast
3730qed
3731
3732AOT_act_theorem "block-paradox2:1": ‹∀x [G]x → ¬[λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3733proof(rule "→I"; rule "raa-cor:2")
3734 AOT_assume antecedant: ‹∀x [G]x›
3735 AOT_have Lemma: ‹∀x ([G]❙ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
3736 proof(rule GEN)
3737 fix x
3738 AOT_have A: ‹[G]❙ιy (y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3739 proof(rule "≡I"; rule "→I")
3740 AOT_assume ‹[G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))›
3741 AOT_hence ‹❙ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
3742 using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
3743 AOT_thus ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3744 using "1-exists:1"[THEN "≡E"(1)] by blast
3745 next
3746 AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3747 AOT_obtain a where a_1: ‹a = x & ∃H (x[H] & ¬[H]x)› and a_2: ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = a)›
3748 using "uniqueness:1"[THEN "≡⇩d⇩fE", OF A] "&E" "∃E"[rotated] by blast
3749 AOT_have a_3: ‹[G]a›
3750 using antecedant "∀E" by blast
3751 AOT_show ‹[G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))›
3752 apply (rule "russell-axiom[exe,1].russell-axiom"[THEN "≡E"(2)])
3753 apply (rule "∃I"(2))
3754 using a_1 a_2 a_3 "&I" by blast
3755 qed
3756 also AOT_have B: ‹... ≡ ∃H (x[H] & ¬[H]x)›
3757 proof (rule "≡I"; rule "→I")
3758 AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3759 AOT_obtain a where ‹a = x & ∃H (x[H] & ¬[H]x)›
3760 using "uniqueness:1"[THEN "≡⇩d⇩fE", OF A] "&E" "∃E"[rotated] by blast
3761 AOT_thus ‹∃H (x[H] & ¬[H]x)› using "&E" by blast
3762 next
3763 AOT_assume ‹∃H (x[H] & ¬[H]x)›
3764 AOT_hence ‹x = x & ∃H (x[H] & ¬[H]x)›
3765 using "id-eq:1" "&I" by blast
3766 moreover AOT_have ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = x)›
3767 by (simp add: "Conjunction Simplification"(1) "universal-cor")
3768 ultimately AOT_show ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3769 using "uniqueness:1"[THEN "≡⇩d⇩fI"] "&I" "∃I"(2) by fast
3770 qed
3771 finally AOT_show ‹([G]❙ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))› .
3772 qed
3773
3774 AOT_assume A: ‹[λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3775 AOT_have θ: ‹∀x ([λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ [G]❙ιy(y = x & ∃H (x[H] & ¬[H]x)))›
3776 using "beta-C-meta"[THEN "→E", OF A] "∀I" by fast
3777 AOT_have ‹∀x ([λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ ∃H (x[H] & ¬[H]x))›
3778 using θ Lemma "cqt-basic:10"[THEN "→E"] "&I" by fast
3779 AOT_hence ‹∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))›
3780 using "∃I"(1) A by fast
3781 AOT_thus ‹(∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))) & (¬∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x)))›
3782 using "block-paradox:2" "&I" by blast
3783qed
3784
3785AOT_act_theorem "block-paradox2:2": ‹∃G ¬[λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3786proof(rule "∃I"(1))
3787 AOT_have 0: ‹[λx ∀p (p →p)]↓›
3788 by "cqt:2[lambda]"
3789 moreover AOT_have ‹∀x [λx ∀p (p →p)]x›
3790 apply (rule GEN)
3791 apply (rule "beta-C-cor:2"[THEN "→E", OF 0, THEN "∀E"(2), THEN "≡E"(2)])
3792 using "if-p-then-p" GEN by fast
3793 moreover AOT_have ‹∀G (∀x [G]x → ¬[λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓)›
3794 using "block-paradox2:1" "∀I" by fast
3795 ultimately AOT_show ‹¬[λx [λx ∀p (p →p)]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3796 using "∀E"(1) "→E" by blast
3797qed("cqt:2[lambda]")
3798
3799AOT_theorem propositions: ‹∃p □(p ≡ φ)›
3800proof(rule "∃I"(1))
3801 AOT_show ‹□(φ ≡ φ)›
3802 by (simp add: RN "oth-class-taut:3:a")
3803next
3804 AOT_show ‹φ↓›
3805 by (simp add: "log-prop-prop:2")
3806qed
3807
3808AOT_theorem "pos-not-equiv-ne:1": ‹(◇¬∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)) → F ≠ G›
3809proof (rule "→I")
3810 AOT_assume ‹◇¬∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)›
3811 AOT_hence ‹¬□∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)›
3812 using "KBasic:11"[THEN "≡E"(2)] by blast
3813 AOT_hence ‹¬(F = G)›
3814 using "id-rel-nec-equiv:1" "modus-tollens:1" by blast
3815 AOT_thus ‹F ≠ G›
3816 using "=-infix"[THEN "≡⇩d⇩fI"] by blast
3817qed
3818
3819AOT_theorem "pos-not-equiv-ne:2": ‹(◇¬(φ{F} ≡ φ{G})) → F ≠ G›
3820proof (rule "→I")
3821 AOT_modally_strict {
3822 AOT_have ‹¬(φ{F} ≡ φ{G}) → ¬(F = G)›
3823 proof (rule "→I"; rule "raa-cor:2")
3824 AOT_assume 1: ‹F = G›
3825 AOT_hence ‹φ{F} → φ{G}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3826 moreover {
3827 AOT_have ‹G = F› using 1 id_sym by blast
3828 AOT_hence ‹φ{G} → φ{F}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3829 }
3830 ultimately AOT_have ‹φ{F} ≡ φ{G}› using "≡I" by blast
3831 moreover AOT_assume ‹¬(φ{F} ≡ φ{G})›
3832 ultimately AOT_show ‹(φ{F} ≡ φ{G}) & ¬(φ{F} ≡ φ{G})›
3833 using "&I" by blast
3834 qed
3835 }
3836 AOT_hence ‹◇¬(φ{F} ≡ φ{G}) → ◇¬(F = G)›
3837 using "RM:2[prem]" by blast
3838 moreover AOT_assume ‹◇¬(φ{F} ≡ φ{G})›
3839 ultimately AOT_have 0: ‹◇¬(F = G)› using "→E" by blast
3840 AOT_have ‹◇(F ≠ G)›
3841 by (AOT_subst "«F ≠ G»" "«¬(F = G)»")
3842 (auto simp: "=-infix" "≡Df" 0)
3843 AOT_thus ‹F ≠ G›
3844 using "id-nec2:3"[THEN "→E"] by blast
3845qed
3846
3847AOT_theorem "pos-not-equiv-ne:2[zero]": ‹(◇¬(φ{p} ≡ φ{q})) → p ≠ q›
3848proof (rule "→I")
3849 AOT_modally_strict {
3850 AOT_have ‹¬(φ{p} ≡ φ{q}) → ¬(p = q)›
3851 proof (rule "→I"; rule "raa-cor:2")
3852 AOT_assume 1: ‹p = q›
3853 AOT_hence ‹φ{p} → φ{q}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3854 moreover {
3855 AOT_have ‹q = p› using 1 id_sym by blast
3856 AOT_hence ‹φ{q} → φ{p}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3857 }
3858 ultimately AOT_have ‹φ{p} ≡ φ{q}› using "≡I" by blast
3859 moreover AOT_assume ‹¬(φ{p} ≡ φ{q})›
3860 ultimately AOT_show ‹(φ{p} ≡ φ{q}) & ¬(φ{p} ≡ φ{q})›
3861 using "&I" by blast
3862 qed
3863 }
3864 AOT_hence ‹◇¬(φ{p} ≡ φ{q}) → ◇¬(p = q)›
3865 using "RM:2[prem]" by blast
3866 moreover AOT_assume ‹◇¬(φ{p} ≡ φ{q})›
3867 ultimately AOT_have 0: ‹◇¬(p = q)› using "→E" by blast
3868 AOT_have ‹◇(p ≠ q)›
3869 by (AOT_subst "«p ≠ q»" "«¬(p = q)»")
3870 (auto simp: 0 "=-infix" "≡Df")
3871 AOT_thus ‹p ≠ q›
3872 using "id-nec2:3"[THEN "→E"] by blast
3873qed
3874
3875AOT_theorem "pos-not-equiv-ne:3": ‹(¬∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)) → F ≠ G›
3876 using "→I" "pos-not-equiv-ne:1"[THEN "→E"] "T◇"[THEN "→E"] by blast
3877
3878AOT_theorem "pos-not-equiv-ne:4": ‹(¬(φ{F} ≡ φ{G})) → F ≠ G›
3879 using "→I" "pos-not-equiv-ne:2"[THEN "→E"] "T◇"[THEN "→E"] by blast
3880
3881AOT_theorem "pos-not-equiv-ne:4[zero]": ‹(¬(φ{p} ≡ φ{q})) → p ≠ q›
3882 using "→I" "pos-not-equiv-ne:2[zero]"[THEN "→E"] "T◇"[THEN "→E"] by blast
3883
3884AOT_define relation_negation :: "Π ⇒ Π" ("_⇧-")
3885 "df-relation-negation": "[F]⇧- =⇩d⇩f [λx⇩1...x⇩n ¬[F]x⇩1...x⇩n]"
3886
3887nonterminal φneg
3888syntax "" :: "φneg ⇒ τ" ("_")
3889syntax "" :: "φneg ⇒ φ" ("'(_')")
3890
3891AOT_define relation_negation_0 :: ‹φ ⇒ φneg› ("'(_')⇧-")
3892 "df-relation-negation[zero]": "(p)⇧- =⇩d⇩f [λ ¬p]"
3893
3894AOT_theorem "rel-neg-T:1": ‹[λx⇩1...x⇩n ¬[Π]x⇩1...x⇩n]↓›
3895 by "cqt:2[lambda]"
3896
3897AOT_theorem "rel-neg-T:1[zero]": ‹[λ ¬φ]↓›
3898 using "cqt:2[lambda0]"[axiom_inst] by blast
3899
3900AOT_theorem "rel-neg-T:2": ‹[Π]⇧- = [λx⇩1...x⇩n ¬[Π]x⇩1...x⇩n]›
3901 using "=I"(1)[OF "rel-neg-T:1"]
3902 by (rule "=⇩d⇩fI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"])
3903
3904AOT_theorem "rel-neg-T:2[zero]": ‹(φ)⇧- = [λ ¬φ]›
3905 using "=I"(1)[OF "rel-neg-T:1[zero]"]
3906 by (rule "=⇩d⇩fI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"])
3907
3908AOT_theorem "rel-neg-T:3": ‹[Π]⇧-↓›
3909 using "=⇩d⇩fI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"] "rel-neg-T:1" by blast
3910
3911AOT_theorem "rel-neg-T:3[zero]": ‹(φ)⇧-↓›
3912 using "log-prop-prop:2" by blast
3913
3914
3915
3916AOT_theorem "thm-relation-negation:1": ‹[F]⇧-x⇩1...x⇩n ≡ ¬[F]x⇩1...x⇩n›
3917proof -
3918 AOT_have ‹[F]⇧-x⇩1...x⇩n ≡ [λx⇩1...x⇩n ¬[F]x⇩1...x⇩n]x⇩1...x⇩n›
3919 using "rule=E"[rotated, OF "rel-neg-T:2"] "rule=E"[rotated, OF "rel-neg-T:2"[THEN id_sym]]
3920 "→I" "≡I" by fast
3921 also AOT_have ‹... ≡ ¬[F]x⇩1...x⇩n›
3922 using "beta-C-meta"[THEN "→E", OF "rel-neg-T:1"] by fast
3923 finally show ?thesis.
3924qed
3925
3926AOT_theorem "thm-relation-negation:2": ‹¬[F]⇧-x⇩1...x⇩n ≡ [F]x⇩1...x⇩n›
3927 apply (AOT_subst "«[F]x⇩1...x⇩n»" "«¬¬[F]x⇩1...x⇩n»")
3928 apply (simp add: "oth-class-taut:3:b")
3929 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
3930 using "thm-relation-negation:1".
3931
3932AOT_theorem "thm-relation-negation:3": ‹((p)⇧-) ≡ ¬p›
3933proof -
3934 AOT_have ‹(p)⇧- = [λ ¬p]› using "rel-neg-T:2[zero]" by blast
3935 AOT_hence ‹((p)⇧-) ≡ [λ ¬p]›
3936 using "df-relation-negation[zero]" "log-prop-prop:2" "oth-class-taut:3:a" "rule-id-def:2:a" by blast
3937 also AOT_have ‹[λ ¬p] ≡ ¬p›
3938 by (simp add: "propositions-lemma:2")
3939 finally show ?thesis.
3940qed
3941
3942AOT_theorem "thm-relation-negation:4": ‹(¬((p)⇧-)) ≡ p›
3943 using "thm-relation-negation:3"[THEN "≡E"(1)]
3944 "thm-relation-negation:3"[THEN "≡E"(2)]
3945 "≡I" "→I" RAA by metis
3946
3947AOT_theorem "thm-relation-negation:5": ‹[F] ≠ [F]⇧-›
3948proof -
3949 AOT_have ‹¬([F] = [F]⇧-)›
3950 proof (rule RAA(2))
3951 AOT_show ‹[F]x⇩1...x⇩n → [F]x⇩1...x⇩n› for x⇩1x⇩n
3952 using "if-p-then-p".
3953 next
3954 AOT_assume ‹[F] = [F]⇧-›
3955 AOT_hence ‹[F]⇧- = [F]› using id_sym by blast
3956 AOT_hence ‹[F]x⇩1...x⇩n ≡ ¬[F]x⇩1...x⇩n› for x⇩1x⇩n
3957 using "rule=E" "thm-relation-negation:1" by fast
3958 AOT_thus ‹¬([F]x⇩1...x⇩n → [F]x⇩1...x⇩n)› for x⇩1x⇩n
3959 using "≡E" RAA by metis
3960 qed
3961 thus ?thesis
3962 using "≡⇩d⇩fI" "=-infix" by blast
3963qed
3964
3965AOT_theorem "thm-relation-negation:6": ‹p ≠ (p)⇧-›
3966proof -
3967 AOT_have ‹¬(p = (p)⇧-)›
3968 proof (rule RAA(2))
3969 AOT_show ‹p → p›
3970 using "if-p-then-p".
3971 next
3972 AOT_assume ‹p = (p)⇧-›
3973 AOT_hence ‹(p)⇧- = p› using id_sym by blast
3974 AOT_hence ‹p ≡ ¬p›
3975 using "rule=E" "thm-relation-negation:3" by fast
3976 AOT_thus ‹¬(p → p)›
3977 using "≡E" RAA by metis
3978 qed
3979 thus ?thesis
3980 using "≡⇩d⇩fI" "=-infix" by blast
3981qed
3982
3983AOT_theorem "thm-relation-negation:7": ‹(p)⇧- = (¬p)›
3984 apply (rule "df-relation-negation[zero]"[THEN "=⇩d⇩fE"(1)])
3985 using "cqt:2[lambda0]"[axiom_inst] "rel-neg-T:2[zero]" "propositions-lemma:1" id_trans by blast+
3986
3987AOT_theorem "thm-relation-negation:8": ‹p = q → (¬p) = (¬q)›
3988proof(rule "→I")
3989 AOT_assume ‹p = q›
3990 moreover AOT_have ‹(¬p)↓› using "log-prop-prop:2".
3991 moreover AOT_have ‹(¬p) = (¬p)› using calculation(2) "=I" by blast
3992 ultimately AOT_show ‹(¬p) = (¬q)›
3993 using "rule=E" by fast
3994qed
3995
3996AOT_theorem "thm-relation-negation:9": ‹p = q → (p)⇧- = (q)⇧-›
3997proof(rule "→I")
3998 AOT_assume ‹p = q›
3999 AOT_hence ‹(¬p) = (¬q)› using "thm-relation-negation:8" "→E" by blast
4000 AOT_thus ‹(p)⇧- = (q)⇧-›
4001 using "thm-relation-negation:7" id_sym id_trans by metis
4002qed
4003
4004AOT_define Necessary :: ‹Π ⇒ φ› ("Necessary'(_')")
4005 "contingent-properties:1": ‹Necessary([F]) ≡⇩d⇩f □∀x⇩1...∀x⇩n [F]x⇩1...x⇩n›
4006
4007AOT_define Necessary0 :: ‹φ ⇒ φ› ("Necessary0'(_')")
4008 "contingent-properties:1[zero]": ‹Necessary0(p) ≡⇩d⇩f □p›
4009
4010AOT_define Impossible :: ‹Π ⇒ φ› ("Impossible'(_')")
4011 "contingent-properties:2": ‹Impossible([F]) ≡⇩d⇩f F↓ & □∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n›
4012
4013AOT_define Impossible0 :: ‹φ ⇒ φ› ("Impossible0'(_')")
4014 "contingent-properties:2[zero]": ‹Impossible0(p) ≡⇩d⇩f □¬p›
4015
4016AOT_define NonContingent :: ‹Π ⇒ φ› ("NonContingent'(_')")
4017 "contingent-properties:3": ‹NonContingent([F]) ≡⇩d⇩f Necessary([F]) ∨ Impossible([F])›
4018
4019AOT_define NonContingent0 :: ‹φ ⇒ φ› ("NonContingent0'(_')")
4020 "contingent-properties:3[zero]": ‹NonContingent0(p) ≡⇩d⇩f Necessary0(p) ∨ Impossible0(p)›
4021
4022AOT_define Contingent :: ‹Π ⇒ φ› ("Contingent'(_')")
4023 "contingent-properties:4": ‹Contingent([F]) ≡⇩d⇩f F↓ & ¬(Necessary([F]) ∨ Impossible([F]))›
4024
4025AOT_define Contingent0 :: ‹φ ⇒ φ› ("Contingent0'(_')")
4026 "contingent-properties:4[zero]": ‹Contingent0(p) ≡⇩d⇩f ¬(Necessary0(p) ∨ Impossible0(p))›
4027
4028
4029AOT_theorem "thm-cont-prop:1": ‹NonContingent([F]) ≡ NonContingent([F]⇧-)›
4030proof (rule "≡I"; rule "→I")
4031 AOT_assume ‹NonContingent([F])›
4032 AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4033 using "≡⇩d⇩fE"[OF "contingent-properties:3"] by blast
4034 moreover {
4035 AOT_assume ‹Necessary([F])›
4036 AOT_hence ‹□(∀x⇩1...∀x⇩n [F]x⇩1...x⇩n)›
4037 using "≡⇩d⇩fE"[OF "contingent-properties:1"] by blast
4038 moreover AOT_modally_strict {
4039 AOT_assume ‹∀x⇩1...∀x⇩n [F]x⇩1...x⇩n›
4040 AOT_hence ‹[F]x⇩1...x⇩n› for x⇩1x⇩n using "∀E" by blast
4041 AOT_hence ‹¬[F]⇧-x⇩1...x⇩n› for x⇩1x⇩n
4042 by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:2" "≡E"(1))
4043 AOT_hence ‹∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n› using "∀I" by fast
4044 }
4045 ultimately AOT_have ‹□(∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n)›
4046 using "RN[prem]"[where Γ="{«∀x⇩1...∀x⇩n [F]x⇩1...x⇩n»}", simplified] by blast
4047 AOT_hence ‹Impossible([F]⇧-)›
4048 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(2)]
4049 by blast
4050 }
4051 moreover {
4052 AOT_assume ‹Impossible([F])›
4053 AOT_hence ‹□(∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n)›
4054 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)]
4055 by blast
4056 moreover AOT_modally_strict {
4057 AOT_assume ‹∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n›
4058 AOT_hence ‹¬[F]x⇩1...x⇩n› for x⇩1x⇩n using "∀E" by blast
4059 AOT_hence ‹[F]⇧-x⇩1...x⇩n› for x⇩1x⇩n
4060 by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(1))
4061 AOT_hence ‹∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n› using "∀I" by fast
4062 }
4063 ultimately AOT_have ‹□(∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n)›
4064 using "RN[prem]"[where Γ="{«∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n»}"] by blast
4065 AOT_hence ‹Necessary([F]⇧-)›
4066 using "≡⇩d⇩fI"[OF "contingent-properties:1"] by blast
4067 }
4068 ultimately AOT_have ‹Necessary([F]⇧-) ∨ Impossible([F]⇧-)›
4069 using "∨E"(1) "∨I" "→I" by metis
4070 AOT_thus ‹NonContingent([F]⇧-)›
4071 using "≡⇩d⇩fI"[OF "contingent-properties:3"] by blast
4072next
4073 AOT_assume ‹NonContingent([F]⇧-)›
4074 AOT_hence ‹Necessary([F]⇧-) ∨ Impossible([F]⇧-)›
4075 using "≡⇩d⇩fE"[OF "contingent-properties:3"] by blast
4076 moreover {
4077 AOT_assume ‹Necessary([F]⇧-)›
4078 AOT_hence ‹□(∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n)›
4079 using "≡⇩d⇩fE"[OF "contingent-properties:1"] by blast
4080 moreover AOT_modally_strict {
4081 AOT_assume ‹∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n›
4082 AOT_hence ‹[F]⇧-x⇩1...x⇩n› for x⇩1x⇩n using "∀E" by blast
4083 AOT_hence ‹¬[F]x⇩1...x⇩n› for x⇩1x⇩n
4084 by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(2))
4085 AOT_hence ‹∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n› using "∀I" by fast
4086 }
4087 ultimately AOT_have ‹□∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n›
4088 using "RN[prem]"[where Γ="{«∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n»}"] by blast
4089 AOT_hence ‹Impossible([F])›
4090 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(2)]
4091 by blast
4092 }
4093 moreover {
4094 AOT_assume ‹Impossible([F]⇧-)›
4095 AOT_hence ‹□(∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n)›
4096 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(1)]
4097 by blast
4098 moreover AOT_modally_strict {
4099 AOT_assume ‹∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n›
4100 AOT_hence ‹¬[F]⇧-x⇩1...x⇩n› for x⇩1x⇩n using "∀E" by blast
4101 AOT_hence ‹[F]x⇩1...x⇩n› for x⇩1x⇩n
4102 using "thm-relation-negation:1"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4103 "useful-tautologies:1"[THEN "→E"] by blast
4104 AOT_hence ‹∀x⇩1...∀x⇩n [F]x⇩1...x⇩n› using "∀I" by fast
4105 }
4106 ultimately AOT_have ‹□(∀x⇩1...∀x⇩n [F]x⇩1...x⇩n)›
4107 using "RN[prem]"[where Γ="{«∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n»}"] by blast
4108 AOT_hence ‹Necessary([F])›
4109 using "≡⇩d⇩fI"[OF "contingent-properties:1"] by blast
4110 }
4111 ultimately AOT_have ‹Necessary([F]) ∨ Impossible([F])›
4112 using "∨E"(1) "∨I" "→I" by metis
4113 AOT_thus ‹NonContingent([F])›
4114 using "≡⇩d⇩fI"[OF "contingent-properties:3"] by blast
4115qed
4116
4117AOT_theorem "thm-cont-prop:2": ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4118proof -
4119 AOT_have ‹Contingent([F]) ≡ ¬(Necessary([F]) ∨ Impossible([F]))›
4120 using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]]
4121 by blast
4122 also AOT_have ‹... ≡ ¬Necessary([F]) & ¬Impossible([F])›
4123 using "oth-class-taut:5:d" by fastforce
4124 also AOT_have ‹... ≡ ¬Impossible([F]) & ¬Necessary([F])›
4125 by (simp add: "Commutativity of &")
4126 also AOT_have ‹... ≡ ◇∃x [F]x & ¬Necessary([F])›
4127 proof (rule "oth-class-taut:4:e"[THEN "→E"])
4128 AOT_have ‹¬Impossible([F]) ≡ ¬□¬ ∃x [F]x›
4129 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4130 apply (AOT_subst "«∃x [F]x»" "«¬ ∀x ¬[F]x»")
4131 apply (simp add: "conventions:4" "≡Df")
4132 apply (AOT_subst_rev "«∀x ¬[F]x»" "«¬¬∀x ¬[F]x»" )
4133 apply (simp add: "oth-class-taut:3:b")
4134 using "contingent-properties:2"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]] by blast
4135 also AOT_have ‹... ≡ ◇∃x [F]x›
4136 using "conventions:5"[THEN "≡Df", symmetric] by blast
4137 finally AOT_show ‹¬Impossible([F]) ≡ ◇∃x [F]x› .
4138 qed
4139 also AOT_have ‹... ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4140 proof (rule "oth-class-taut:4:f"[THEN "→E"])
4141 AOT_have ‹¬Necessary([F]) ≡ ¬□¬∃x ¬[F]x›
4142 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4143 apply (AOT_subst "«∃x ¬[F]x»" "«¬ ∀x ¬¬[F]x»")
4144 apply (simp add: "conventions:4" "≡Df")
4145 apply (AOT_subst_rev "λ κ . «[F]κ»" "λ κ . «¬¬[F]κ»")
4146 apply (simp add: "oth-class-taut:3:b")
4147 apply (AOT_subst_rev "«∀x [F]x»" "«¬¬∀x [F]x»")
4148 by (auto simp: "oth-class-taut:3:b" "contingent-properties:1" "≡Df")
4149 also AOT_have ‹... ≡ ◇∃x ¬[F]x›
4150 using "conventions:5"[THEN "≡Df", symmetric] by blast
4151 finally AOT_show ‹¬Necessary([F]) ≡ ◇∃x ¬[F]x›.
4152 qed
4153 finally show ?thesis.
4154qed
4155
4156AOT_theorem "thm-cont-prop:3": ‹Contingent([F]) ≡ Contingent([F]⇧-)› for F::‹<κ> AOT_var›
4157proof -
4158 {
4159 fix Π :: ‹<κ>›
4160 AOT_assume ‹Π↓›
4161 moreover AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4162 using "thm-cont-prop:2" GEN by fast
4163 ultimately AOT_have ‹Contingent([Π]) ≡ ◇∃x [Π]x & ◇∃x ¬[Π]x›
4164 using "thm-cont-prop:2" "∀E" by fast
4165 } note 1 = this
4166 AOT_have ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4167 using "thm-cont-prop:2" by blast
4168 also AOT_have ‹... ≡ ◇∃x ¬[F]x & ◇∃x [F]x›
4169 by (simp add: "Commutativity of &")
4170 also AOT_have ‹... ≡ ◇∃x [F]⇧-x & ◇∃x [F]x›
4171 by (AOT_subst "λ κ . «[F]⇧-κ»" "λκ . «¬[F]κ»")
4172 (auto simp: "thm-relation-negation:1" "oth-class-taut:3:a")
4173 also AOT_have ‹... ≡ ◇∃x [F]⇧-x & ◇∃x ¬[F]⇧-x›
4174 by (AOT_subst_rev "λ κ . «¬[F]⇧-κ»" "λκ . «[F]κ»")
4175 (auto simp: "thm-relation-negation:2" "oth-class-taut:3:a")
4176 also AOT_have ‹... ≡ Contingent([F]⇧-)›
4177 using 1[OF "rel-neg-T:3", symmetric] by blast
4178 finally show ?thesis.
4179qed
4180
4181AOT_define concrete_if_concrete :: ‹Π› ("L") L_def: ‹L =⇩d⇩f [λx E!x → E!x]›
4182
4183AOT_theorem "thm-noncont-e-e:1": ‹Necessary(L)›
4184proof -
4185 AOT_modally_strict {
4186 fix x
4187 AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4188 moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4189 moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4190 ultimately AOT_have ‹[λx E!x → E!x]x›
4191 using "β←C" by blast
4192 }
4193 AOT_hence 0: ‹□∀x [λx E!x → E!x]x›
4194 using RN GEN by blast
4195 show ?thesis
4196 apply (rule "=⇩d⇩fI"(2)[OF L_def])
4197 apply "cqt:2[lambda]"
4198 by (rule "contingent-properties:1"[THEN "≡⇩d⇩fI", OF 0])
4199qed
4200
4201AOT_theorem "thm-noncont-e-e:2": ‹Impossible([L]⇧-)›
4202proof -
4203 AOT_modally_strict {
4204 fix x
4205
4206 AOT_have 0: ‹∀F (¬[F]⇧-x ≡ [F]x)›
4207 using "thm-relation-negation:2" GEN by fast
4208 AOT_have ‹¬[λx E!x → E!x]⇧-x ≡ [λx E!x → E!x]x›
4209 by (rule 0[THEN "∀E"(1)]) "cqt:2[lambda]"
4210 moreover {
4211 AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4212 moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4213 moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4214 ultimately AOT_have ‹[λx E!x → E!x]x›
4215 using "β←C" by blast
4216 }
4217 ultimately AOT_have ‹¬[λx E!x → E!x]⇧-x›
4218 using "≡E" by blast
4219 }
4220 AOT_hence 0: ‹□∀x ¬[λx E!x → E!x]⇧-x›
4221 using RN GEN by fast
4222 show ?thesis
4223 apply (rule "=⇩d⇩fI"(2)[OF L_def])
4224 apply "cqt:2[lambda]"
4225 apply (rule "contingent-properties:2"[THEN "≡⇩d⇩fI"]; rule "&I")
4226 using "rel-neg-T:3"
4227 apply blast
4228 using 0
4229 by blast
4230qed
4231
4232AOT_theorem "thm-noncont-e-e:3": ‹NonContingent(L)›
4233 using "thm-noncont-e-e:1"
4234 by (rule "contingent-properties:3"[THEN "≡⇩d⇩fI", OF "∨I"(1)])
4235
4236AOT_theorem "thm-noncont-e-e:4": ‹NonContingent([L]⇧-)›
4237proof -
4238 AOT_have 0: ‹∀F (NonContingent([F]) ≡ NonContingent([F]⇧-))›
4239 using "thm-cont-prop:1" "∀I" by fast
4240 moreover AOT_have 1: ‹L↓›
4241 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
4242 AOT_show ‹NonContingent([L]⇧-)›
4243 using "∀E"(1)[OF 0, OF 1, THEN "≡E"(1), OF "thm-noncont-e-e:3"] by blast
4244qed
4245
4246AOT_theorem "thm-noncont-e-e:5": ‹∃F ∃G (F ≠ «G::<κ>» & NonContingent([F]) & NonContingent([G]))›
4247proof (rule "∃I")+
4248 {
4249 AOT_have ‹∀F [F] ≠ [F]⇧-› using "thm-relation-negation:5" GEN by fast
4250 moreover AOT_have ‹L↓›
4251 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
4252 ultimately AOT_have ‹L ≠ [L]⇧-› using "∀E" by blast
4253 }
4254 AOT_thus ‹L ≠ [L]⇧- & NonContingent(L) & NonContingent([L]⇧-)›
4255 using "thm-noncont-e-e:3" "thm-noncont-e-e:4" "&I" by metis
4256next
4257 AOT_show ‹[L]⇧-↓›
4258 using "rel-neg-T:3" by blast
4259next
4260 AOT_show ‹L↓›
4261 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
4262qed
4263
4264AOT_theorem "lem-cont-e:1": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4265proof -
4266 AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ∃x ◇([F]x & ◇¬[F]x)›
4267 using "BF◇" "CBF◇" "≡I" by blast
4268 also AOT_have ‹… ≡ ∃x (◇[F]x & ◇¬[F]x)›
4269 by (AOT_subst ‹λκ. «◇([F]κ & ◇¬[F]κ)»› ‹λ κ . «◇[F]κ & ◇¬[F]κ»›)
4270 (auto simp: "S5Basic:11" "cqt-further:7")
4271 also AOT_have ‹… ≡ ∃x (◇¬[F]x & ◇[F]x)›
4272 by (AOT_subst ‹λκ. «◇¬[F]κ & ◇[F]κ»› ‹λ κ . «◇[F]κ & ◇¬[F]κ»›)
4273 (auto simp: "Commutativity of &" "cqt-further:7")
4274 also AOT_have ‹… ≡ ∃x ◇(¬[F]x & ◇[F]x)›
4275 by (AOT_subst ‹λ κ . «◇(¬[F]κ & ◇[F]κ)»› ‹λκ. «◇¬[F]κ & ◇[F]κ»›)
4276 (auto simp: "S5Basic:11" "oth-class-taut:3:a")
4277 also AOT_have ‹… ≡ ◇∃x (¬[F]x & ◇[F]x)›
4278 using "BF◇" "CBF◇" "≡I" by fast
4279 finally show ?thesis.
4280qed
4281
4282AOT_theorem "lem-cont-e:2": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x ([F]⇧-x & ◇¬[F]⇧-x)›
4283proof -
4284 AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4285 using "lem-cont-e:1".
4286 also AOT_have ‹… ≡ ◇∃x ([F]⇧-x & ◇¬[F]⇧-x)›
4287 apply (AOT_subst "λ κ . «¬[F]⇧-κ»" "λ κ . «[F]κ»")
4288 apply (simp add: "thm-relation-negation:2")
4289 apply (AOT_subst "λ κ . «[F]⇧-κ»" "λ κ . «¬[F]κ»")
4290 apply (simp add: "thm-relation-negation:1")
4291 by (simp add: "oth-class-taut:3:a")
4292 finally show ?thesis.
4293qed
4294
4295AOT_theorem "thm-cont-e:1": ‹◇∃x (E!x & ◇¬E!x)›
4296proof (rule "CBF◇"[THEN "→E"])
4297 AOT_have ‹∃x ◇(E!x & ¬❙𝒜E!x)› using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] by blast
4298 then AOT_obtain a where ‹◇(E!a & ¬❙𝒜E!a)› using "∃E"[rotated] by blast
4299 AOT_hence θ: ‹◇E!a & ◇¬❙𝒜E!a›
4300 using "KBasic2:3"[THEN "→E"] by blast
4301 AOT_have ξ: ‹◇E!a & ◇❙𝒜¬E!a›
4302 by (AOT_subst "«❙𝒜¬E!a»" "«¬❙𝒜E!a»")
4303 (auto simp: "logic-actual-nec:1"[axiom_inst] θ)
4304 AOT_have ζ: ‹◇E!a & ❙𝒜¬E!a›
4305 by (AOT_subst "«❙𝒜¬E!a»" "«◇❙𝒜¬E!a»")
4306 (auto simp add: "Act-Sub:4" ξ)
4307 AOT_hence ‹◇E!a & ◇¬E!a›
4308 using "&E" "&I" "Act-Sub:3"[THEN "→E"] by blast
4309 AOT_hence ‹◇(E!a & ◇¬E!a)› using "S5Basic:11"[THEN "≡E"(2)] by simp
4310 AOT_thus ‹∃x ◇(E!x & ◇¬E!x)› using "∃I"(2) by fast
4311qed
4312
4313AOT_theorem "thm-cont-e:2": ‹◇∃x (¬E!x & ◇E!x)›
4314proof -
4315 AOT_have ‹∀F (◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x))›
4316 using "lem-cont-e:1" GEN by fast
4317 AOT_hence ‹(◇∃x (E!x & ◇¬E!x) ≡ ◇∃x (¬E!x & ◇E!x))›
4318 using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4319 thus ?thesis using "thm-cont-e:1" "≡E" by blast
4320qed
4321
4322AOT_theorem "thm-cont-e:3": ‹◇∃x E!x›
4323proof (rule "CBF◇"[THEN "→E"])
4324 AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4325 using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4326 AOT_hence ‹◇E!a›
4327 using "KBasic2:3"[THEN "→E", THEN "&E"(1)] by blast
4328 AOT_thus ‹∃x ◇E!x› using "∃I" by fast
4329qed
4330
4331AOT_theorem "thm-cont-e:4": ‹◇∃x ¬E!x›
4332proof (rule "CBF◇"[THEN "→E"])
4333 AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4334 using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4335 AOT_hence ‹◇◇¬E!a›
4336 using "KBasic2:3"[THEN "→E", THEN "&E"(2)] by blast
4337 AOT_hence ‹◇¬E!a›
4338 using "4◇"[THEN "→E"] by blast
4339 AOT_thus ‹∃x ◇¬E!x› using "∃I" by fast
4340qed
4341
4342AOT_theorem "thm-cont-e:5": ‹Contingent([E!])›
4343proof -
4344 AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4345 using "thm-cont-prop:2" GEN by fast
4346 AOT_hence ‹Contingent([E!]) ≡ ◇∃x E!x & ◇∃x ¬E!x›
4347 using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4348 thus ?thesis
4349 using "thm-cont-e:3" "thm-cont-e:4" "≡E"(2) "&I" by blast
4350qed
4351
4352AOT_theorem "thm-cont-e:6": ‹Contingent([E!]⇧-)›
4353proof -
4354 AOT_have ‹∀F (Contingent([«F::<κ>»]) ≡ Contingent([F]⇧-))›
4355 using "thm-cont-prop:3" GEN by fast
4356 AOT_hence ‹Contingent([E!]) ≡ Contingent([E!]⇧-)›
4357 using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4358 thus ?thesis using "thm-cont-e:5" "≡E" by blast
4359qed
4360
4361AOT_theorem "thm-cont-e:7": ‹∃F∃G (Contingent([«F::<κ>»]) & Contingent([G]) & F ≠ G)›
4362proof (rule "∃I")+
4363 AOT_have ‹∀F [«F::<κ>»] ≠ [F]⇧-› using "thm-relation-negation:5" GEN by fast
4364 AOT_hence ‹[E!] ≠ [E!]⇧-›
4365 using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4366 AOT_thus ‹Contingent([E!]) & Contingent([E!]⇧-) & [E!] ≠ [E!]⇧-›
4367 using "thm-cont-e:5" "thm-cont-e:6" "&I" by metis
4368next
4369 AOT_show ‹E!⇧-↓›
4370 by (fact AOT)
4371next
4372 AOT_show ‹E!↓› by (fact "cqt:2[concrete]"[axiom_inst])
4373qed
4374
4375AOT_theorem "property-facts:1": ‹NonContingent([F]) → ¬∃G (Contingent([G]) & G = F)›
4376proof (rule "→I"; rule "raa-cor:2")
4377 AOT_assume ‹NonContingent([F])›
4378 AOT_hence 1: ‹Necessary([F]) ∨ Impossible([F])›
4379 using "contingent-properties:3"[THEN "≡⇩d⇩fE"] by blast
4380 AOT_assume ‹∃G (Contingent([G]) & G = F)›
4381 then AOT_obtain G where ‹Contingent([G]) & G = F› using "∃E"[rotated] by blast
4382 AOT_hence ‹Contingent([F])› using "rule=E" "&E" by blast
4383 AOT_hence ‹¬(Necessary([F]) ∨ Impossible([F]))›
4384 using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4385 AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4386 using 1 "&I" by blast
4387qed
4388
4389AOT_theorem "property-facts:2": ‹Contingent([F]) → ¬∃G (NonContingent([G]) & G = F)›
4390proof (rule "→I"; rule "raa-cor:2")
4391 AOT_assume ‹Contingent([F])›
4392 AOT_hence 1: ‹¬(Necessary([F]) ∨ Impossible([F]))›
4393 using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4394 AOT_assume ‹∃G (NonContingent([G]) & G = F)›
4395 then AOT_obtain G where ‹NonContingent([G]) & G = F› using "∃E"[rotated] by blast
4396 AOT_hence ‹NonContingent([F])› using "rule=E" "&E" by blast
4397 AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4398 using "contingent-properties:3"[THEN "≡⇩d⇩fE"] by blast
4399 AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4400 using 1 "&I" by blast
4401qed
4402
4403AOT_theorem "property-facts:3": ‹L ≠ [L]⇧- & L ≠ E! & L ≠ E!⇧- & [L]⇧- ≠ [E!]⇧- & E! ≠ [E!]⇧-›
4404proof -
4405 AOT_have noneqI: ‹Π ≠ Π'› if ‹φ{Π}› and ‹¬φ{Π'}› for φ Π Π'
4406 apply (rule "=-infix"[THEN "≡⇩d⇩fI"]; rule "raa-cor:2")
4407 using "rule=E"[where φ=φ and τ=Π and σ = Π'] that "&I" by blast
4408 AOT_have contingent_denotes: ‹Π↓› if ‹Contingent([Π])› for Π
4409 using that "contingent-properties:4"[THEN "≡⇩d⇩fE", THEN "&E"(1)] by blast
4410 AOT_have not_noncontingent_if_contingent: ‹¬NonContingent([Π])› if ‹Contingent([Π])› for Π
4411 proof(rule RAA(2))
4412 AOT_show ‹¬(Necessary([Π]) ∨ Impossible([Π]))›
4413 using that "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF contingent_denotes[OF that], THEN "≡E"(1)] by blast
4414 next
4415 AOT_assume ‹NonContingent([Π])›
4416 AOT_thus ‹Necessary([Π]) ∨ Impossible([Π])›
4417 using "contingent-properties:3"[THEN "≡⇩d⇩fE"] by blast
4418 qed
4419
4420 show ?thesis
4421 proof (rule "&I")+
4422 AOT_show ‹L ≠ [L]⇧-›
4423 apply (rule "=⇩d⇩fI"(2)[OF L_def])
4424 apply "cqt:2[lambda]"
4425 apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]⇧-»"])
4426 apply (rule GEN) apply (fact AOT)
4427 by "cqt:2[lambda]"
4428 next
4429 AOT_show ‹L ≠ E!›
4430 apply (rule noneqI)
4431 using "thm-noncont-e-e:3" not_noncontingent_if_contingent[OF "thm-cont-e:5"]
4432 by auto
4433 next
4434 AOT_show ‹L ≠ E!⇧-›
4435 apply (rule noneqI)
4436 using "thm-noncont-e-e:3" apply fast
4437 apply (rule not_noncontingent_if_contingent)
4438 apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]⇧-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4439 using "thm-cont-prop:3" GEN apply fast
4440 using "thm-cont-e:5" by fast+
4441 next
4442 AOT_show ‹[L]⇧- ≠ E!⇧-›
4443 apply (rule noneqI)
4444 using "thm-noncont-e-e:4" apply fast
4445 apply (rule not_noncontingent_if_contingent)
4446 apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]⇧-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4447 using "thm-cont-prop:3" GEN apply fast
4448 using "thm-cont-e:5" by fast+
4449 next
4450 AOT_show ‹E! ≠ E!⇧-›
4451 apply (rule "=⇩d⇩fI"(2)[OF L_def])
4452 apply "cqt:2[lambda]"
4453 apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]⇧-»"])
4454 apply (rule GEN) apply (fact AOT)
4455 by (fact "cqt:2[concrete]"[axiom_inst])
4456 qed
4457qed
4458
4459AOT_theorem "thm-cont-propos:1": ‹NonContingent0(p) ≡ NonContingent0(((p)⇧-))›
4460proof(rule "≡I"; rule "→I")
4461 AOT_assume ‹NonContingent0(p)›
4462 AOT_hence ‹Necessary0(p) ∨ Impossible0(p)›
4463 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fE"] by blast
4464 moreover {
4465 AOT_assume ‹Necessary0(p)›
4466 AOT_hence 1: ‹□p› using "contingent-properties:1[zero]"[THEN "≡⇩d⇩fE"] by blast
4467 AOT_have ‹□¬((p)⇧-)›
4468 by (AOT_subst "«¬((p)⇧-)»" "AOT_term_of_var p")
4469 (auto simp add: 1 "thm-relation-negation:4")
4470 AOT_hence ‹Impossible0(((p)⇧-))›
4471 by (rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fI"])
4472 }
4473 moreover {
4474 AOT_assume ‹Impossible0(p)›
4475 AOT_hence 1: ‹□¬p›
4476 by (rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fE"])
4477 AOT_have ‹□((p)⇧-)›
4478 by (AOT_subst "«((p)⇧-)»" "«¬p»")
4479 (auto simp: 1 "thm-relation-negation:3")
4480 AOT_hence ‹Necessary0(((p)⇧-))›
4481 by (rule "contingent-properties:1[zero]"[THEN "≡⇩d⇩fI"])
4482 }
4483 ultimately AOT_have ‹Necessary0(((p)⇧-)) ∨ Impossible0(((p)⇧-))›
4484 using "∨E"(1) "∨I" "→I" by metis
4485 AOT_thus ‹NonContingent0(((p)⇧-))›
4486 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fI"] by blast
4487next
4488 AOT_assume ‹NonContingent0(((p)⇧-))›
4489 AOT_hence ‹Necessary0(((p)⇧-)) ∨ Impossible0(((p)⇧-))›
4490 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fE"] by blast
4491 moreover {
4492 AOT_assume ‹Impossible0(((p)⇧-))›
4493 AOT_hence 1: ‹□¬((p)⇧-)›
4494 by (rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fE"])
4495 AOT_have ‹□p›
4496 by (AOT_subst_rev "«¬((p)⇧-)»" "AOT_term_of_var p")
4497 (auto simp: 1 "thm-relation-negation:4")
4498 AOT_hence ‹Necessary0(p)›
4499 using "contingent-properties:1[zero]"[THEN "≡⇩d⇩fI"] by blast
4500 }
4501 moreover {
4502 AOT_assume ‹Necessary0(((p)⇧-))›
4503 AOT_hence 1: ‹□((p)⇧-)›
4504 by (rule "contingent-properties:1[zero]"[THEN "≡⇩d⇩fE"])
4505 AOT_have ‹□¬p›
4506 by (AOT_subst_rev "«((p)⇧-)»" "«¬p»")
4507 (auto simp: 1 "thm-relation-negation:3")
4508 AOT_hence ‹Impossible0(p)›
4509 by (rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fI"])
4510 }
4511 ultimately AOT_have ‹Necessary0(p) ∨ Impossible0(p)›
4512 using "∨E"(1) "∨I" "→I" by metis
4513 AOT_thus ‹NonContingent0(p)›
4514 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fI"] by blast
4515qed
4516
4517AOT_theorem "thm-cont-propos:2": ‹Contingent0(φ) ≡ ◇φ & ◇¬φ›
4518proof -
4519 AOT_have ‹Contingent0(φ) ≡ ¬(Necessary0(φ) ∨ Impossible0(φ))›
4520 using "contingent-properties:4[zero]"[THEN "≡Df"] by simp
4521 also AOT_have ‹… ≡ ¬Necessary0(φ) & ¬Impossible0(φ)›
4522 by (fact AOT)
4523 also AOT_have ‹… ≡ ¬Impossible0(φ) & ¬Necessary0(φ)›
4524 by (fact AOT)
4525 also AOT_have ‹… ≡ ◇φ & ◇¬φ›
4526 apply (AOT_subst "«◇φ»" "«¬□¬φ»")
4527 apply (simp add: "conventions:5" "≡Df")
4528 apply (AOT_subst "«Impossible0(φ)»" "«□¬φ»")
4529 apply (simp add: "contingent-properties:2[zero]" "≡Df")
4530 apply (AOT_subst_rev "«¬□φ»" "«◇¬φ»")
4531 apply (simp add: "KBasic:11")
4532 apply (AOT_subst "«Necessary0(φ)»" "«□φ»")
4533 apply (simp add: "contingent-properties:1[zero]" "≡Df")
4534 by (simp add: "oth-class-taut:3:a")
4535 finally show ?thesis.
4536qed
4537
4538AOT_theorem "thm-cont-propos:3": ‹Contingent0(p) ≡ Contingent0(((p)⇧-))›
4539proof -
4540 AOT_have ‹Contingent0(p) ≡ ◇p & ◇¬p› using "thm-cont-propos:2".
4541 also AOT_have ‹… ≡ ◇¬p & ◇p› by (fact AOT)
4542 also AOT_have ‹… ≡ ◇((p)⇧-) & ◇p›
4543 by (AOT_subst "«((p)⇧-)»" "«¬p»")
4544 (auto simp: "thm-relation-negation:3" "oth-class-taut:3:a")
4545 also AOT_have ‹… ≡ ◇((p)⇧-) & ◇¬((p)⇧-)›
4546 by (AOT_subst "«¬((p)⇧-)»" "AOT_term_of_var p")
4547 (auto simp: "thm-relation-negation:4" "oth-class-taut:3:a")
4548 also AOT_have ‹… ≡ Contingent0(((p)⇧-))›
4549 using "thm-cont-propos:2"[symmetric] by blast
4550 finally show ?thesis.
4551qed
4552
4553AOT_define noncontingent_prop :: ‹φ› ("p⇩0")
4554 p⇩0_def: "(p⇩0) =⇩d⇩f (∀x (E!x → E!x))"
4555
4556AOT_theorem "thm-noncont-propos:1": ‹Necessary0((p⇩0))›
4557proof(rule "contingent-properties:1[zero]"[THEN "≡⇩d⇩fI"])
4558 AOT_show ‹□(p⇩0)›
4559 apply (rule "=⇩d⇩fI"(2)[OF p⇩0_def])
4560 using "log-prop-prop:2" apply simp
4561 using "if-p-then-p" RN GEN by fast
4562qed
4563
4564AOT_theorem "thm-noncont-propos:2": ‹Impossible0(((p⇩0)⇧-))›
4565proof(rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fI"])
4566 AOT_show ‹□¬((p⇩0)⇧-)›
4567 apply (AOT_subst "«((p⇩0)⇧-)»" "«¬p⇩0»")
4568 using "thm-relation-negation:3" GEN "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4569 apply (AOT_subst_rev "«p⇩0»" "«¬¬p⇩0»" )
4570 apply (simp add: "oth-class-taut:3:b")
4571 apply (rule "=⇩d⇩fI"(2)[OF p⇩0_def])
4572 using "log-prop-prop:2" apply simp
4573 using "if-p-then-p" RN GEN by fast
4574qed
4575
4576AOT_theorem "thm-noncont-propos:3": ‹NonContingent0((p⇩0))›
4577 apply(rule "contingent-properties:3[zero]"[THEN "≡⇩d⇩fI"])
4578 using "thm-noncont-propos:1" "∨I" by blast
4579
4580AOT_theorem "thm-noncont-propos:4": ‹NonContingent0(((p⇩0)⇧-))›
4581 apply(rule "contingent-properties:3[zero]"[THEN "≡⇩d⇩fI"])
4582 using "thm-noncont-propos:2" "∨I" by blast
4583
4584AOT_theorem "thm-noncont-propos:5": ‹∃p∃q (NonContingent0((p)) & NonContingent0((q)) & p ≠ q)›
4585proof(rule "∃I")+
4586 AOT_have 0: ‹φ ≠ (φ)⇧-› for φ
4587 using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4588 AOT_thus ‹NonContingent0((p⇩0)) & NonContingent0(((p⇩0)⇧-)) & (p⇩0) ≠ (p⇩0)⇧-›
4589 using "thm-noncont-propos:3" "thm-noncont-propos:4" "&I" by auto
4590qed(auto simp: "log-prop-prop:2")
4591
4592AOT_act_theorem "no-cnac": ‹¬∃x(E!x & ¬❙𝒜E!x)›
4593proof(rule "raa-cor:2")
4594 AOT_assume ‹∃x(E!x & ¬❙𝒜E!x)›
4595 then AOT_obtain a where a: ‹E!a & ¬❙𝒜E!a›
4596 using "∃E"[rotated] by blast
4597 AOT_hence ‹❙𝒜¬E!a› using "&E" "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
4598 AOT_hence ‹¬E!a› using "logic-actual"[act_axiom_inst, THEN "→E"] by blast
4599 AOT_hence ‹E!a & ¬E!a› using a "&E" "&I" by blast
4600 AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
4601qed
4602
4603AOT_theorem "pos-not-pna:1": ‹¬❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4604proof(rule "raa-cor:2")
4605 AOT_assume ‹❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4606 AOT_hence ‹∃x ❙𝒜(E!x & ¬❙𝒜E!x)›
4607 using "Act-Basic:10"[THEN "≡E"(1)] by blast
4608 then AOT_obtain a where ‹❙𝒜(E!a & ¬❙𝒜E!a)› using "∃E"[rotated] by blast
4609 AOT_hence 1: ‹❙𝒜E!a & ❙𝒜¬❙𝒜E!a› using "Act-Basic:2"[THEN "≡E"(1)] by blast
4610 AOT_hence ‹¬❙𝒜❙𝒜E!a› using "&E"(2) "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by blast
4611 AOT_hence ‹¬❙𝒜E!a› using "logic-actual-nec:4"[axiom_inst, THEN "≡E"(1)] RAA by blast
4612 AOT_thus ‹p & ¬p› for p using 1[THEN "&E"(1)] "&I" "raa-cor:1" by blast
4613qed
4614
4615AOT_theorem "pos-not-pna:2": ‹◇¬∃x(E!x & ¬❙𝒜E!x)›
4616proof (rule RAA(1))
4617 AOT_show ‹¬❙𝒜∃x (E!x & ¬❙𝒜E!x)› using "pos-not-pna:1" by blast
4618next
4619 AOT_assume ‹¬◇¬∃x (E!x & ¬❙𝒜E!x)›
4620 AOT_hence ‹□∃x (E!x & ¬❙𝒜E!x)›
4621 using "KBasic:12"[THEN "≡E"(2)] by blast
4622 AOT_thus ‹❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4623 using "nec-imp-act"[THEN "→E"] by blast
4624qed
4625
4626AOT_theorem "pos-not-pna:3": ‹∃x (◇E!x & ¬❙𝒜E!x)›
4627proof -
4628 AOT_obtain a where ‹◇(E!a & ¬❙𝒜E!a)›
4629 using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] "∃E"[rotated] by blast
4630 AOT_hence θ: ‹◇E!a› and ξ: ‹◇¬❙𝒜E!a› using "KBasic2:3"[THEN "→E"] "&E" by blast+
4631 AOT_have ‹¬□❙𝒜E!a› using ξ "KBasic:11"[THEN "≡E"(2)] by blast
4632 AOT_hence ‹¬❙𝒜E!a› using "Act-Basic:6"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)] by blast
4633 AOT_hence ‹◇E!a & ¬❙𝒜E!a› using θ "&I" by blast
4634 thus ?thesis using "∃I" by fast
4635qed
4636
4637AOT_define contingent_prop :: φ ("q⇩0")
4638 q⇩0_def: ‹(q⇩0) =⇩d⇩f (∃x (E!x & ¬❙𝒜E!x))›
4639
4640AOT_theorem q⇩0_prop: ‹◇q⇩0 & ◇¬q⇩0›
4641 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
4642 apply (fact "log-prop-prop:2")
4643 apply (rule "&I")
4644 apply (fact "qml:4"[axiom_inst])
4645 by (fact "pos-not-pna:2")
4646
4647AOT_theorem "basic-prop:1": ‹Contingent0((q⇩0))›
4648proof(rule "contingent-properties:4[zero]"[THEN "≡⇩d⇩fI"])
4649 AOT_have ‹¬Necessary0((q⇩0)) & ¬Impossible0((q⇩0))›
4650 proof (rule "&I"; rule "=⇩d⇩fI"(2)[OF q⇩0_def]; (rule "log-prop-prop:2" | rule "raa-cor:2"))
4651 AOT_assume ‹Necessary0(∃x (E!x & ¬❙𝒜E!x))›
4652 AOT_hence ‹□∃x (E!x & ¬❙𝒜E!x)›
4653 using "contingent-properties:1[zero]"[THEN "≡⇩d⇩fE"] by blast
4654 AOT_hence ‹❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4655 using "Act-Basic:8"[THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
4656 AOT_thus ‹❙𝒜∃x (E!x & ¬❙𝒜E!x) & ¬❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4657 using "pos-not-pna:1" "&I" by blast
4658 next
4659 AOT_assume ‹Impossible0(∃x (E!x & ¬❙𝒜E!x))›
4660 AOT_hence ‹□¬(∃x (E!x & ¬❙𝒜E!x))›
4661 using "contingent-properties:2[zero]"[THEN "≡⇩d⇩fE"] by blast
4662 AOT_hence ‹¬◇(∃x (E!x & ¬❙𝒜E!x))› using "KBasic2:1"[THEN "≡E"(1)] by blast
4663 AOT_thus ‹◇(∃x (E!x & ¬❙𝒜E!x)) & ¬◇(∃x (E!x & ¬❙𝒜E!x))›
4664 using "qml:4"[axiom_inst] "&I" by blast
4665 qed
4666 AOT_thus ‹¬(Necessary0((q⇩0)) ∨ Impossible0((q⇩0)))›
4667 using "oth-class-taut:5:d" "≡E"(2) by blast
4668qed
4669
4670AOT_theorem "basic-prop:2": ‹∃p Contingent0((p))›
4671 using "∃I"(1)[rotated, OF "log-prop-prop:2"] "basic-prop:1" by blast
4672
4673AOT_theorem "basic-prop:3": ‹Contingent0(((q⇩0)⇧-))›
4674 apply (AOT_subst "«(q⇩0)⇧-»" "«¬q⇩0»")
4675 apply (insert "thm-relation-negation:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"]; fast)
4676 apply (rule "contingent-properties:4[zero]"[THEN "≡⇩d⇩fI"])
4677 apply (rule "oth-class-taut:5:d"[THEN "≡E"(2)])
4678 apply (rule "&I")
4679 apply (rule "contingent-properties:1[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4680 apply (rule "conventions:5"[THEN "≡⇩d⇩fE"])
4681 apply (rule "=⇩d⇩fE"(2)[OF q⇩0_def])
4682 apply (rule "log-prop-prop:2")
4683 apply (rule q⇩0_prop[THEN "&E"(1)])
4684 apply (rule "contingent-properties:2[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4685 apply (rule "conventions:5"[THEN "≡⇩d⇩fE"])
4686 by (rule q⇩0_prop[THEN "&E"(2)])
4687
4688AOT_theorem "basic-prop:4": ‹∃p∃q (p ≠ q & Contingent0(p) & Contingent0(q))›
4689proof(rule "∃I")+
4690 AOT_have 0: ‹φ ≠ (φ)⇧-› for φ
4691 using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4692 AOT_show ‹(q⇩0) ≠ (q⇩0)⇧- & Contingent0(q⇩0) & Contingent0(((q⇩0)⇧-))›
4693 using "basic-prop:1" "basic-prop:3" "&I" 0 by presburger
4694qed(auto simp: "log-prop-prop:2")
4695
4696AOT_theorem "proposition-facts:1": ‹NonContingent0(p) → ¬∃q (Contingent0(q) & q = p)›
4697proof(rule "→I"; rule "raa-cor:2")
4698 AOT_assume ‹NonContingent0(p)›
4699 AOT_hence 1: ‹Necessary0(p) ∨ Impossible0(p)›
4700 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fE"] by blast
4701 AOT_assume ‹∃q (Contingent0(q) & q = p)›
4702 then AOT_obtain q where ‹Contingent0(q) & q = p› using "∃E"[rotated] by blast
4703 AOT_hence ‹Contingent0(p)› using "rule=E" "&E" by fast
4704 AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4705 using "contingent-properties:4[zero]"[THEN "≡⇩d⇩fE"] 1 "&I" by blast
4706qed
4707
4708AOT_theorem "proposition-facts:2": ‹Contingent0(p) → ¬∃q (NonContingent0(q) & q = p)›
4709proof(rule "→I"; rule "raa-cor:2")
4710 AOT_assume ‹Contingent0(p)›
4711 AOT_hence 1: ‹¬(Necessary0(p) ∨ Impossible0(p))›
4712 using "contingent-properties:4[zero]"[THEN "≡⇩d⇩fE"] by blast
4713 AOT_assume ‹∃q (NonContingent0(q) & q = p)›
4714 then AOT_obtain q where ‹NonContingent0(q) & q = p› using "∃E"[rotated] by blast
4715 AOT_hence ‹NonContingent0(p)› using "rule=E" "&E" by fast
4716 AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4717 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fE"] 1 "&I" by blast
4718qed
4719
4720AOT_theorem "proposition-facts:3": ‹(p⇩0) ≠ (p⇩0)⇧- & (p⇩0) ≠ (q⇩0) & (p⇩0) ≠ (q⇩0)⇧- & (p⇩0)⇧- ≠ (q⇩0)⇧- & (q⇩0) ≠ (q⇩0)⇧-›
4721proof -
4722 {
4723 fix χ φ ψ
4724 AOT_assume ‹χ{φ}›
4725 moreover AOT_assume ‹¬χ{ψ}›
4726 ultimately AOT_have ‹¬(χ{φ} ≡ χ{ψ})›
4727 using RAA "≡E" by metis
4728 moreover {
4729 AOT_have ‹∀p∀q ((¬(χ{p} ≡ χ{q})) → p ≠ q)›
4730 by (rule "∀I"; rule "∀I"; rule "pos-not-equiv-ne:4[zero]")
4731 AOT_hence ‹((¬(χ{φ} ≡ χ{ψ})) → φ ≠ ψ)›
4732 using "∀E" "log-prop-prop:2" by blast
4733 }
4734 ultimately AOT_have ‹φ ≠ ψ›
4735 using "→E" by blast
4736 } note 0 = this
4737 AOT_have contingent_neg: ‹Contingent0(φ) ≡ Contingent0(((φ)⇧-))› for φ
4738 using "thm-cont-propos:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4739 AOT_have not_noncontingent_if_contingent: ‹¬NonContingent0(φ)› if ‹Contingent0(φ)› for φ
4740 apply (rule "contingent-properties:3[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4741 using that "contingent-properties:4[zero]"[THEN "≡⇩d⇩fE"] by blast
4742 show ?thesis
4743 apply (rule "&I")+
4744 using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4745 apply (rule 0)
4746 using "thm-noncont-propos:3" apply fast
4747 apply (rule not_noncontingent_if_contingent)
4748 apply (fact AOT)
4749 apply (rule 0)
4750 apply (rule "thm-noncont-propos:3")
4751 apply (rule not_noncontingent_if_contingent)
4752 apply (rule contingent_neg[THEN "≡E"(1)])
4753 apply (fact AOT)
4754 apply (rule 0)
4755 apply (rule "thm-noncont-propos:4")
4756 apply (rule not_noncontingent_if_contingent)
4757 apply (rule contingent_neg[THEN "≡E"(1)])
4758 apply (fact AOT)
4759 using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4760qed
4761
4762AOT_define "cont-tf:1" :: ‹φ ⇒ φ› ("ContingentlyTrue'(_')")
4763 "cont-tf:1": ‹ContingentlyTrue(p) ≡⇩d⇩f p & ◇¬p›
4764
4765AOT_define "cont-tf:2" :: ‹φ ⇒ φ› ("ContingentlyFalse'(_')")
4766 "cont-tf:2": ‹ContingentlyFalse(p) ≡⇩d⇩f ¬p & ◇p›
4767
4768AOT_theorem "cont-true-cont:1": ‹ContingentlyTrue((p)) → Contingent0((p))›
4769proof(rule "→I")
4770 AOT_assume ‹ContingentlyTrue((p))›
4771 AOT_hence 1: ‹p› and 2: ‹◇¬p› using "cont-tf:1"[THEN "≡⇩d⇩fE"] "&E" by blast+
4772 AOT_have ‹¬Necessary0((p))›
4773 apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4774 using 2 "KBasic:11"[THEN "≡E"(2)] by blast
4775 moreover AOT_have ‹¬Impossible0((p))›
4776 apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4777 apply (rule "conventions:5"[THEN "≡⇩d⇩fE"])
4778 using "T◇"[THEN "→E", OF 1].
4779 ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4780 using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4781 AOT_thus ‹Contingent0((p))›
4782 using "contingent-properties:4[zero]"[THEN "≡⇩d⇩fI"] by blast
4783qed
4784
4785AOT_theorem "cont-true-cont:2": ‹ContingentlyFalse((p)) → Contingent0((p))›
4786proof(rule "→I")
4787 AOT_assume ‹ContingentlyFalse((p))›
4788 AOT_hence 1: ‹¬p› and 2: ‹◇p› using "cont-tf:2"[THEN "≡⇩d⇩fE"] "&E" by blast+
4789 AOT_have ‹¬Necessary0((p))›
4790 apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4791 using "KBasic:11"[THEN "≡E"(2)] "T◇"[THEN "→E", OF 1] by blast
4792 moreover AOT_have ‹¬Impossible0((p))›
4793 apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4794 apply (rule "conventions:5"[THEN "≡⇩d⇩fE"])
4795 using 2.
4796 ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4797 using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4798 AOT_thus ‹Contingent0((p))›
4799 using "contingent-properties:4[zero]"[THEN "≡⇩d⇩fI"] by blast
4800qed
4801
4802AOT_theorem "cont-true-cont:3": ‹ContingentlyTrue((p)) ≡ ContingentlyFalse(((p)⇧-))›
4803proof(rule "≡I"; rule "→I")
4804 AOT_assume ‹ContingentlyTrue((p))›
4805 AOT_hence 0: ‹p & ◇¬p› using "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
4806 AOT_have 1: ‹ContingentlyFalse(¬p)›
4807 apply (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
4808 apply (AOT_subst_rev "AOT_term_of_var p" "«¬¬p»")
4809 by (auto simp: "oth-class-taut:3:b" 0)
4810 AOT_show ‹ContingentlyFalse(((p)⇧-))›
4811 apply (AOT_subst "«(p)⇧-»" "«¬p»")
4812 by (auto simp: "thm-relation-negation:3" 1)
4813next
4814 AOT_assume 1: ‹ContingentlyFalse(((p)⇧-))›
4815 AOT_have ‹ContingentlyFalse(¬p)›
4816 by (AOT_subst_rev "«(p)⇧-»" "«¬p»")
4817 (auto simp: "thm-relation-negation:3" 1)
4818 AOT_hence ‹¬¬p & ◇¬p› using "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
4819 AOT_hence ‹p & ◇¬p›
4820 using "&I" "&E" "useful-tautologies:1"[THEN "→E"] by metis
4821 AOT_thus ‹ContingentlyTrue((p))›
4822 using "cont-tf:1"[THEN "≡⇩d⇩fI"] by blast
4823qed
4824
4825AOT_theorem "cont-true-cont:4": ‹ContingentlyFalse((p)) ≡ ContingentlyTrue(((p)⇧-))›
4826proof(rule "≡I"; rule "→I")
4827 AOT_assume ‹ContingentlyFalse(p)›
4828 AOT_hence 0: ‹¬p & ◇p›
4829 using "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
4830 AOT_have ‹¬p & ◇¬¬p›
4831 by (AOT_subst_rev "AOT_term_of_var p" "«¬¬p»")
4832 (auto simp: "oth-class-taut:3:b" 0)
4833 AOT_hence 1: ‹ContingentlyTrue(¬p)›
4834 by (rule "cont-tf:1"[THEN "≡⇩d⇩fI"])
4835 AOT_show ‹ContingentlyTrue(((p)⇧-))›
4836 by (AOT_subst "«(p)⇧-»" "«¬p»")
4837 (auto simp: "thm-relation-negation:3" 1)
4838next
4839 AOT_assume 1: ‹ContingentlyTrue(((p)⇧-))›
4840 AOT_have ‹ContingentlyTrue(¬p)›
4841 by (AOT_subst_rev "«(p)⇧-»" "«¬p»")
4842 (auto simp add: "thm-relation-negation:3" 1)
4843 AOT_hence 2: ‹¬p & ◇¬¬p› using "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
4844 AOT_have ‹◇p›
4845 by (AOT_subst "AOT_term_of_var p" "«¬¬p»")
4846 (auto simp add: "oth-class-taut:3:b" 2[THEN "&E"(2)])
4847 AOT_hence ‹¬p & ◇p› using 2[THEN "&E"(1)] "&I" by blast
4848 AOT_thus ‹ContingentlyFalse(p)›
4849 by (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
4850qed
4851
4852AOT_theorem "cont-true-cont:5": ‹(ContingentlyTrue((p)) & Necessary0((q))) → p ≠ q›
4853proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4854 AOT_assume ‹ContingentlyTrue((p))›
4855 AOT_hence ‹◇¬p›
4856 using "cont-tf:1"[THEN "≡⇩d⇩fE"] "&E" by blast
4857 AOT_hence 0: ‹¬□p› using "KBasic:11"[THEN "≡E"(2)] by blast
4858 AOT_assume ‹Necessary0((q))›
4859 moreover AOT_assume ‹¬(p ≠ q)›
4860 AOT_hence ‹p = q›
4861 using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4862 "useful-tautologies:1"[THEN "→E"] by blast
4863 ultimately AOT_have ‹Necessary0((p))› using "rule=E" id_sym by blast
4864 AOT_hence ‹□p›
4865 using "contingent-properties:1[zero]"[THEN "≡⇩d⇩fE"] by blast
4866 AOT_thus ‹□p & ¬□p› using 0 "&I" by blast
4867qed
4868
4869AOT_theorem "cont-true-cont:6": ‹(ContingentlyFalse((p)) & Impossible0((q))) → p ≠ q›
4870proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4871 AOT_assume ‹ContingentlyFalse((p))›
4872 AOT_hence ‹◇p›
4873 using "cont-tf:2"[THEN "≡⇩d⇩fE"] "&E" by blast
4874 AOT_hence 1: ‹¬□¬p›
4875 using "conventions:5"[THEN "≡⇩d⇩fE"] by blast
4876 AOT_assume ‹Impossible0((q))›
4877 moreover AOT_assume ‹¬(p ≠ q)›
4878 AOT_hence ‹p = q›
4879 using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4880 "useful-tautologies:1"[THEN "→E"] by blast
4881 ultimately AOT_have ‹Impossible0((p))› using "rule=E" id_sym by blast
4882 AOT_hence ‹□¬p›
4883 using "contingent-properties:2[zero]"[THEN "≡⇩d⇩fE"] by blast
4884 AOT_thus ‹□¬p & ¬□¬p› using 1 "&I" by blast
4885qed
4886
4887AOT_act_theorem "q0cf:1": ‹ContingentlyFalse(q⇩0)›
4888 apply (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
4889 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
4890 apply (fact "log-prop-prop:2")
4891 apply (rule "&I")
4892 apply (fact "no-cnac")
4893 by (fact "qml:4"[axiom_inst])
4894
4895AOT_act_theorem "q0cf:2": ‹ContingentlyTrue(((q⇩0)⇧-))›
4896 apply (rule "cont-tf:1"[THEN "≡⇩d⇩fI"])
4897 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
4898 apply (fact "log-prop-prop:2")
4899 apply (rule "&I")
4900 apply (rule "thm-relation-negation:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(2)])
4901 apply (fact "no-cnac")
4902 apply (rule "rule=E"[rotated, OF "thm-relation-negation:7"[unvarify p, OF "log-prop-prop:2", THEN id_sym]])
4903 apply (AOT_subst_rev "«∃x (E!x & ¬❙𝒜E!x)»" "«¬¬(∃x (E!x & ¬❙𝒜E!x))»")
4904 by (auto simp: "oth-class-taut:3:b" "qml:4"[axiom_inst])
4905
4906
4907
4908AOT_theorem "cont-tf-thm:1": ‹∃p ContingentlyTrue((p))›
4909proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
4910 AOT_assume ‹q⇩0›
4911 AOT_hence ‹q⇩0 & ◇¬q⇩0› using q⇩0_prop[THEN "&E"(2)] "&I" by blast
4912 AOT_thus ‹ContingentlyTrue(q⇩0)›
4913 by (rule "cont-tf:1"[THEN "≡⇩d⇩fI"])
4914next
4915 AOT_assume ‹¬q⇩0›
4916 AOT_hence ‹¬q⇩0 & ◇q⇩0› using q⇩0_prop[THEN "&E"(1)] "&I" by blast
4917 AOT_hence ‹ContingentlyFalse(q⇩0)›
4918 by (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
4919 AOT_thus ‹ContingentlyTrue(((q⇩0)⇧-))›
4920 by (rule "cont-true-cont:4"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
4921qed(auto simp: "log-prop-prop:2")
4922
4923
4924AOT_theorem "cont-tf-thm:2": ‹∃p ContingentlyFalse((p))›
4925proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
4926 AOT_assume ‹q⇩0›
4927 AOT_hence ‹q⇩0 & ◇¬q⇩0› using q⇩0_prop[THEN "&E"(2)] "&I" by blast
4928 AOT_hence ‹ContingentlyTrue(q⇩0)›
4929 by (rule "cont-tf:1"[THEN "≡⇩d⇩fI"])
4930 AOT_thus ‹ContingentlyFalse(((q⇩0)⇧-))›
4931 by (rule "cont-true-cont:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
4932next
4933 AOT_assume ‹¬q⇩0›
4934 AOT_hence ‹¬q⇩0 & ◇q⇩0› using q⇩0_prop[THEN "&E"(1)] "&I" by blast
4935 AOT_thus ‹ContingentlyFalse(q⇩0)›
4936 by (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
4937qed(auto simp: "log-prop-prop:2")
4938
4939
4940AOT_theorem "property-facts1:1": ‹∃F∃x ([F]x & ◇¬[F]x)›
4941proof -
4942 fix x
4943 AOT_obtain p⇩1 where ‹ContingentlyTrue((p⇩1))›
4944 using "cont-tf-thm:1" "∃E"[rotated] by blast
4945 AOT_hence 1: ‹p⇩1 & ◇¬p⇩1› using "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
4946 AOT_modally_strict {
4947 AOT_have ‹for arbitrary p: ❙⊢⇩□ ([λz p]x ≡ p)›
4948 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4949 AOT_hence ‹for arbitrary p: ❙⊢⇩□ □ ([λz p]x ≡ p)›
4950 by (rule RN)
4951 AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
4952 AOT_hence ‹□([λz p⇩1]x ≡ p⇩1)› using "∀E" by fast
4953 } note 2 = this
4954 AOT_hence ‹□([λz p⇩1]x ≡ p⇩1)› using "∀E" by blast
4955 AOT_hence ‹[λz p⇩1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
4956 moreover AOT_have ‹◇¬[λz p⇩1]x›
4957 apply (AOT_subst_using subst: 2[THEN "qml:2"[axiom_inst, THEN "→E"]])
4958 using 1[THEN "&E"(2)] by blast
4959 ultimately AOT_have ‹[λz p⇩1]x & ◇¬[λz p⇩1]x› using "&I" by blast
4960 AOT_hence ‹∃x ([λz p⇩1]x & ◇¬[λz p⇩1]x)› using "∃I"(2) by fast
4961 moreover AOT_have ‹[λz p⇩1]↓› by "cqt:2[lambda]"
4962 ultimately AOT_show ‹∃F∃x ([F]x & ◇¬[F]x)› by (rule "∃I"(1))
4963qed
4964
4965
4966AOT_theorem "property-facts1:2": ‹∃F∃x (¬[F]x & ◇[F]x)›
4967proof -
4968 fix x
4969 AOT_obtain p⇩1 where ‹ContingentlyFalse((p⇩1))›
4970 using "cont-tf-thm:2" "∃E"[rotated] by blast
4971 AOT_hence 1: ‹¬p⇩1 & ◇p⇩1› using "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
4972 AOT_modally_strict {
4973 AOT_have ‹for arbitrary p: ❙⊢⇩□ ([λz p]x ≡ p)›
4974 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4975 AOT_hence ‹for arbitrary p: ❙⊢⇩□ (¬[λz p]x ≡ ¬p)›
4976 using "oth-class-taut:4:b" "≡E" by blast
4977 AOT_hence ‹for arbitrary p: ❙⊢⇩□ □(¬[λz p]x ≡ ¬p)›
4978 by (rule RN)
4979 AOT_hence ‹∀p □(¬[λz p]x ≡ ¬p)› using GEN by fast
4980 AOT_hence ‹□(¬[λz p⇩1]x ≡ ¬p⇩1)› using "∀E" by fast
4981 } note 2 = this
4982 AOT_hence ‹□(¬[λz p⇩1]x ≡ ¬p⇩1)› using "∀E" by blast
4983 AOT_hence 3: ‹¬[λz p⇩1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
4984 AOT_modally_strict {
4985 AOT_have ‹for arbitrary p: ❙⊢⇩□ ([λz p]x ≡ p)›
4986 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
4987 AOT_hence ‹for arbitrary p: ❙⊢⇩□ □([λz p]x ≡ p)›
4988 by (rule RN)
4989 AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
4990 AOT_hence ‹□([λz p⇩1]x ≡ p⇩1)› using "∀E" by fast
4991 } note 4 = this
4992 AOT_have ‹◇[λz p⇩1]x›
4993 apply (AOT_subst_using subst: 4[THEN "qml:2"[axiom_inst, THEN "→E"]])
4994 using 1[THEN "&E"(2)] by blast
4995 AOT_hence ‹¬[λz p⇩1]x & ◇[λz p⇩1]x› using 3 "&I" by blast
4996 AOT_hence ‹∃x (¬[λz p⇩1]x & ◇[λz p⇩1]x)› using "∃I"(2) by fast
4997 moreover AOT_have ‹[λz p⇩1]↓› by "cqt:2[lambda]"
4998 ultimately AOT_show ‹∃F∃x (¬[F]x & ◇[F]x)› by (rule "∃I"(1))
4999qed
5000
5001context
5002begin
5003
5004private AOT_lemma eqnotnec_123_Aux_ζ: ‹[L]x ≡ (E!x → E!x)›
5005 apply (rule "=⇩d⇩fI"(2)[OF L_def])
5006 apply "cqt:2[lambda]"
5007 apply (rule "beta-C-meta"[THEN "→E"])
5008 by "cqt:2[lambda]"
5009
5010private AOT_lemma eqnotnec_123_Aux_ω: ‹[λz φ]x ≡ φ›
5011 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5012
5013private AOT_lemma eqnotnec_123_Aux_θ: ‹φ ≡ ∀x([L]x ≡ [λz φ]x)›
5014proof(rule "≡I"; rule "→I"; (rule "∀I")?)
5015 fix x
5016 AOT_assume 1: ‹φ›
5017 AOT_have ‹[L]x ≡ (E!x → E!x)› using eqnotnec_123_Aux_ζ.
5018 also AOT_have ‹… ≡ φ›
5019 using "if-p-then-p" 1 "≡I" "→I" by simp
5020 also AOT_have ‹… ≡ [λz φ]x›
5021 using "Commutativity of ≡"[THEN "≡E"(1)] eqnotnec_123_Aux_ω by blast
5022 finally AOT_show ‹[L]x ≡ [λz φ]x›.
5023next
5024 fix x
5025 AOT_assume ‹∀x([L]x ≡ [λz φ]x)›
5026 AOT_hence ‹[L]x ≡ [λz φ]x› using "∀E" by blast
5027 also AOT_have ‹… ≡ φ› using eqnotnec_123_Aux_ω.
5028 finally AOT_have ‹φ ≡ [L]x› using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5029 also AOT_have ‹… ≡ E!x → E!x› using eqnotnec_123_Aux_ζ.
5030 finally AOT_show ‹φ› using "≡E" "if-p-then-p" by fast
5031qed
5032private lemmas eqnotnec_123_Aux_ξ = eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5033 THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5034 THEN "RM◇"]
5035private lemmas eqnotnec_123_Aux_ξ' = eqnotnec_123_Aux_θ[THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)], THEN "RM◇"]
5036
5037AOT_theorem "eqnotnec:1": ‹∃F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5038proof-
5039 AOT_obtain p⇩1 where ‹ContingentlyTrue(p⇩1)› using "cont-tf-thm:1" "∃E"[rotated] by blast
5040 AOT_hence ‹p⇩1 & ◇¬p⇩1› using "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
5041 AOT_hence ‹∀x ([L]x ≡ [λz p⇩1]x) & ◇¬∀x([L]x ≡ [λz p⇩1]x)›
5042 apply - apply (rule "&I")
5043 using "&E" eqnotnec_123_Aux_θ[THEN "≡E"(1)] eqnotnec_123_Aux_ξ "→E" by fast+
5044 AOT_hence ‹∃G (∀x([L]x ≡ [G]x) & ◇¬∀x([L]x ≡ [G]x))›
5045 by (rule "∃I") "cqt:2[lambda]"
5046 AOT_thus ‹∃F∃G (∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5047 apply (rule "∃I")
5048 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5049qed
5050
5051AOT_theorem "eqnotnec:2": ‹∃F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5052proof-
5053 AOT_obtain p⇩1 where ‹ContingentlyFalse(p⇩1)› using "cont-tf-thm:2" "∃E"[rotated] by blast
5054 AOT_hence ‹¬p⇩1 & ◇p⇩1› using "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
5055 AOT_hence ‹¬∀x ([L]x ≡ [λz p⇩1]x) & ◇∀x([L]x ≡ [λz p⇩1]x)›
5056 apply - apply (rule "&I")
5057 using "&E" eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)] eqnotnec_123_Aux_ξ' "→E" by fast+
5058 AOT_hence ‹∃G (¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5059 by (rule "∃I") "cqt:2[lambda]"
5060 AOT_thus ‹∃F∃G (¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5061 apply (rule "∃I")
5062 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5063qed
5064
5065AOT_theorem "eqnotnec:3": ‹∃F∃G(❙𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5066proof-
5067 AOT_have ‹¬❙𝒜q⇩0›
5068 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
5069 apply (fact "log-prop-prop:2")
5070 by (fact AOT)
5071 AOT_hence ‹❙𝒜¬q⇩0›
5072 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5073 AOT_hence ‹❙𝒜¬∀x ([L]x ≡ [λz q⇩0]x)›
5074 using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5075 THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5076 THEN "RA[2]", THEN "act-cond"[THEN "→E"], THEN "→E"] by blast
5077 moreover AOT_have ‹◇∀x ([L]x ≡ [λz q⇩0]x)› using eqnotnec_123_Aux_ξ'[THEN "→E"] q⇩0_prop[THEN "&E"(1)] by blast
5078 ultimately AOT_have ‹❙𝒜¬∀x ([L]x ≡ [λz q⇩0]x) & ◇∀x ([L]x ≡ [λz q⇩0]x)› using "&I" by blast
5079 AOT_hence ‹∃G (❙𝒜¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5080 by (rule "∃I") "cqt:2[lambda]"
5081 AOT_thus ‹∃F∃G (❙𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5082 apply (rule "∃I")
5083 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5084qed
5085
5086end
5087
5088
5090AOT_theorem "eqnotnec:4": ‹∀F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5091proof(rule GEN)
5092 fix F
5093
5094 AOT_have Aux_A: ‹❙⊢⇩□ ψ → ∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5095 proof(rule "→I"; rule GEN)
5096 AOT_modally_strict {
5097 fix x
5098 AOT_assume 0: ‹ψ›
5099 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5100 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5101 also AOT_have ‹... ≡ [F]x›
5102 apply (rule "≡I"; rule "→I")
5103 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5104 using 0 "&I" by blast
5105 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5106 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5107 }
5108 qed
5109
5110 AOT_have Aux_B: ‹❙⊢⇩□ ψ → ∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5111 proof (rule "→I"; rule GEN)
5112 AOT_modally_strict {
5113 fix x
5114 AOT_assume 0: ‹ψ›
5115 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5116 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5117 also AOT_have ‹... ≡ [F]x›
5118 apply (rule "≡I"; rule "→I")
5119 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5120 apply (rule "∨I"(1)) using 0 "&I" by blast
5121 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5122 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5123 }
5124 qed
5125
5126 AOT_have Aux_C: ‹❙⊢⇩□ ◇¬ψ → ◇¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5127 proof(rule "RM◇"; rule "→I"; rule "raa-cor:2")
5128 AOT_modally_strict {
5129 AOT_assume 0: ‹¬ψ›
5130 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5131 AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5132 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5133 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5134 moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5135 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5136 ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5137 using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5138 moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5139 ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5140 AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5141 }
5142 qed
5143
5144 AOT_have Aux_D: ‹□∀z ([F]z ≡ [λz [F]z & ψ]z) → (◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5145 proof (rule "→I")
5146 AOT_assume A: ‹□∀z([F]z ≡ [λz [F]z & ψ]z)›
5147 AOT_show ‹◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5148 proof(rule "≡I"; rule "KBasic:13"[THEN "→E"];
5149 rule "RN[prem]"[where Γ="{«∀z([F]z ≡ [λz [F]z & ψ]z)»}", simplified];
5150 (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5151 AOT_modally_strict {
5152 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5153 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5154 AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5155 AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5156 AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5157 AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5158 }
5159 next
5160 AOT_modally_strict {
5161 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5162 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5163 AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5164 AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5165 AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5166 AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5167 }
5168 qed(auto simp: A)
5169 qed
5170
5171 AOT_obtain p⇩1 where p⇩1_prop: ‹p⇩1 & ◇¬p⇩1› using "cont-tf-thm:1" "∃E"[rotated] "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
5172 {
5173 AOT_assume 1: ‹□∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5174 AOT_have 2: ‹∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5175 using Aux_B[THEN "→E", OF p⇩1_prop[THEN "&E"(1)]].
5176 AOT_have ‹◇¬∀x([λz [F]z & p⇩1]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5177 using Aux_C[THEN "→E", OF p⇩1_prop[THEN "&E"(2)]].
5178 AOT_hence 3: ‹◇¬∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5179 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5180 AOT_hence ‹∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)› using 2 "&I" by blast
5181 AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5182 by (rule "∃I"(1)) "cqt:2[lambda]"
5183 }
5184 moreover {
5185 AOT_assume 2: ‹¬□∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5186 AOT_hence ‹◇¬∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5187 using "KBasic:11"[THEN "≡E"(1)] by blast
5188 AOT_hence ‹∀x ([F]x ≡ [λz [F]z & p⇩1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5189 using Aux_A[THEN "→E", OF p⇩1_prop[THEN "&E"(1)]] "&I" by blast
5190 AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5191 by (rule "∃I"(1)) "cqt:2[lambda]"
5192 }
5193 ultimately AOT_show ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5194 using "∨E"(1)[OF "exc-mid"] "→I" by blast
5195qed
5196
5197AOT_theorem "eqnotnec:5": ‹∀F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5198proof(rule GEN)
5199 fix F
5200
5201 AOT_have Aux_A: ‹❙⊢⇩□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5202 proof(rule "RM◇"; rule "→I"; rule GEN)
5203 AOT_modally_strict {
5204 fix x
5205 AOT_assume 0: ‹ψ›
5206 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5207 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5208 also AOT_have ‹... ≡ [F]x›
5209 apply (rule "≡I"; rule "→I")
5210 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5211 using 0 "&I" by blast
5212 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5213 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5214 }
5215 qed
5216
5217 AOT_have Aux_B: ‹❙⊢⇩□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5218 proof (rule "RM◇"; rule "→I"; rule GEN)
5219 AOT_modally_strict {
5220 fix x
5221 AOT_assume 0: ‹ψ›
5222 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5223 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5224 also AOT_have ‹... ≡ [F]x›
5225 apply (rule "≡I"; rule "→I")
5226 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5227 apply (rule "∨I"(1)) using 0 "&I" by blast
5228 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5229 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5230 }
5231 qed
5232
5233 AOT_have Aux_C: ‹❙⊢⇩□ ¬ψ → ¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5234 proof(rule "→I"; rule "raa-cor:2")
5235 AOT_modally_strict {
5236 AOT_assume 0: ‹¬ψ›
5237 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5238 AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5239 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5240 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5241 moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5242 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5243 ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5244 using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5245 moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5246 ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5247 AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5248 }
5249 qed
5250
5251 AOT_have Aux_D: ‹∀z ([F]z ≡ [λz [F]z & ψ]z) → (¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5252 proof (rule "→I"; rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5253 AOT_modally_strict {
5254 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5255 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5256 AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5257 AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5258 AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5259 AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5260 }
5261 next
5262 AOT_modally_strict {
5263 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5264 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5265 AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5266 AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5267 AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5268 AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5269 }
5270 qed
5271
5272 AOT_obtain p⇩1 where p⇩1_prop: ‹¬p⇩1 & ◇p⇩1› using "cont-tf-thm:2" "∃E"[rotated] "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
5273 {
5274 AOT_assume 1: ‹∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5275 AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5276 using Aux_B[THEN "→E", OF p⇩1_prop[THEN "&E"(2)]].
5277 AOT_have ‹¬∀x([λz [F]z & p⇩1]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5278 using Aux_C[THEN "→E", OF p⇩1_prop[THEN "&E"(1)]].
5279 AOT_hence 3: ‹¬∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5280 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5281 AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x) & ◇∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)› using 2 "&I" by blast
5282 AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5283 by (rule "∃I"(1)) "cqt:2[lambda]"
5284 }
5285 moreover {
5286 AOT_assume 2: ‹¬∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5287 AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5288 using "KBasic:11"[THEN "≡E"(1)] by blast
5289 AOT_hence ‹¬∀x ([F]x ≡ [λz [F]z & p⇩1]x) & ◇∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5290 using Aux_A[THEN "→E", OF p⇩1_prop[THEN "&E"(2)]] "&I" by blast
5291 AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5292 by (rule "∃I"(1)) "cqt:2[lambda]"
5293 }
5294 ultimately AOT_show ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5295 using "∨E"(1)[OF "exc-mid"] "→I" by blast
5296qed
5297
5298AOT_theorem "eqnotnec:6": ‹∀F∃G(❙𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5299proof(rule GEN)
5300 fix F
5301
5302 AOT_have Aux_A: ‹❙⊢⇩□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5303 proof(rule "RM◇"; rule "→I"; rule GEN)
5304 AOT_modally_strict {
5305 fix x
5306 AOT_assume 0: ‹ψ›
5307 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5308 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5309 also AOT_have ‹... ≡ [F]x›
5310 apply (rule "≡I"; rule "→I")
5311 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5312 using 0 "&I" by blast
5313 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5314 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5315 }
5316 qed
5317
5318 AOT_have Aux_B: ‹❙⊢⇩□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5319 proof (rule "RM◇"; rule "→I"; rule GEN)
5320 AOT_modally_strict {
5321 fix x
5322 AOT_assume 0: ‹ψ›
5323 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5324 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5325 also AOT_have ‹... ≡ [F]x›
5326 apply (rule "≡I"; rule "→I")
5327 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5328 apply (rule "∨I"(1)) using 0 "&I" by blast
5329 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5330 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5331 }
5332 qed
5333
5334 AOT_have Aux_C: ‹❙⊢⇩□ ❙𝒜¬ψ → ❙𝒜¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5335 proof(rule "act-cond"[THEN "→E"]; rule "RA[2]"; rule "→I"; rule "raa-cor:2")
5336 AOT_modally_strict {
5337 AOT_assume 0: ‹¬ψ›
5338 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5339 AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5340 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5341 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5342 moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5343 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5344 ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5345 using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5346 moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5347 ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5348 AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5349 }
5350 qed
5351
5352 AOT_have Aux_D: ‹❙𝒜∀z ([F]z ≡ [λz [F]z & ψ]z) → (❙𝒜¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ❙𝒜¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5353 proof (rule "→I"; rule "Act-Basic:5"[THEN "≡E"(1)])
5354 AOT_assume ‹❙𝒜∀z ([F]z ≡ [λz [F]z & ψ]z)›
5355 AOT_thus ‹❙𝒜(¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))›
5356 proof (rule "RA[3]"[where Γ="{«∀z ([F]z ≡ [λz [F]z & ψ]z)»}", simplified, rotated])
5357 AOT_modally_strict {
5358 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5359 AOT_thus ‹¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5360 apply -
5361 proof(rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5362 AOT_modally_strict {
5363 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5364 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5365 AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5366 AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5367 AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5368 AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5369 }
5370 next
5371 AOT_modally_strict {
5372 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5373 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5374 AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5375 AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5376 AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5377 AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5378 }
5379 qed
5380 }
5381 qed
5382 qed
5383
5384 AOT_have ‹¬❙𝒜q⇩0›
5385 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
5386 apply (fact "log-prop-prop:2")
5387 by (fact AOT)
5388 AOT_hence q⇩0_prop_1: ‹❙𝒜¬q⇩0›
5389 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5390 {
5391 AOT_assume 1: ‹❙𝒜∀x([F]x ≡ [λz [F]z & q⇩0]x)›
5392 AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x)›
5393 using Aux_B[THEN "→E", OF q⇩0_prop[THEN "&E"(1)]].
5394 AOT_have ‹❙𝒜¬∀x([λz [F]z & q⇩0]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x)›
5395 using Aux_C[THEN "→E", OF q⇩0_prop_1].
5396 AOT_hence 3: ‹❙𝒜¬∀x([F]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x)›
5397 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5398 AOT_hence ‹❙𝒜¬∀x([F]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x) & ◇∀x([F]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x)› using 2 "&I" by blast
5399 AOT_hence ‹∃G (❙𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5400 by (rule "∃I"(1)) "cqt:2[lambda]"
5401 }
5402 moreover {
5403 AOT_assume 2: ‹¬❙𝒜∀x([F]x ≡ [λz [F]z & q⇩0]x)›
5404 AOT_hence ‹❙𝒜¬∀x([F]x ≡ [λz [F]z & q⇩0]x)›
5405 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5406 AOT_hence ‹❙𝒜¬∀x ([F]x ≡ [λz [F]z & q⇩0]x) & ◇∀x([F]x ≡ [λz [F]z & q⇩0]x)›
5407 using Aux_A[THEN "→E", OF q⇩0_prop[THEN "&E"(1)]] "&I" by blast
5408 AOT_hence ‹∃G (❙𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5409 by (rule "∃I"(1)) "cqt:2[lambda]"
5410 }
5411 ultimately AOT_show ‹∃G (❙𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5412 using "∨E"(1)[OF "exc-mid"] "→I" by blast
5413qed
5414
5415AOT_theorem "oa-contingent:1": ‹O! ≠ A!›
5416proof(rule "≡⇩d⇩fI"[OF "=-infix"]; rule "raa-cor:2")
5417 fix x
5418 AOT_assume 1: ‹O! = A!›
5419 AOT_hence ‹[λx ◇E!x] = A!›
5420 by (rule "=⇩d⇩fE"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5421 AOT_hence ‹[λx ◇E!x] = [λx ¬◇E!x]›
5422 by (rule "=⇩d⇩fE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5423 moreover AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
5424 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5425 ultimately AOT_have ‹[λx ¬◇E!x]x ≡ ◇E!x›
5426 using "rule=E" by fast
5427 moreover AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
5428 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5429 ultimately AOT_have ‹◇E!x ≡ ¬◇E!x› using "≡E"(6) "Commutativity of ≡"[THEN "≡E"(1)] by blast
5430 AOT_thus "(◇E!x ≡ ¬◇E!x) & ¬(◇E!x ≡ ¬◇E!x)" using "oth-class-taut:3:c" "&I" by blast
5431qed
5432
5433AOT_theorem "oa-contingent:2": ‹O!x ≡ ¬A!x›
5434proof -
5435 AOT_have ‹O!x ≡ [λx ◇E!x]x›
5436 apply (rule "≡I"; rule "→I")
5437 apply (rule "=⇩d⇩fE"(2)[OF AOT_ordinary])
5438 apply "cqt:2[lambda]"
5439 apply argo
5440 apply (rule "=⇩d⇩fI"(2)[OF AOT_ordinary])
5441 apply "cqt:2[lambda]"
5442 by argo
5443 also AOT_have ‹… ≡ ◇E!x›
5444 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5445 also AOT_have ‹… ≡ ¬¬◇E!x›
5446 using "oth-class-taut:3:b".
5447 also AOT_have ‹… ≡ ¬[λx ¬◇E!x]x›
5448 by (rule "beta-C-meta"[THEN "→E", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric]) "cqt:2[lambda]"
5449 also AOT_have ‹… ≡ ¬A!x›
5450 apply (rule "≡I"; rule "→I")
5451 apply (rule "=⇩d⇩fI"(2)[OF AOT_abstract])
5452 apply "cqt:2[lambda]"
5453 apply argo
5454 apply (rule "=⇩d⇩fE"(2)[OF AOT_abstract])
5455 apply "cqt:2[lambda]"
5456 by argo
5457 finally show ?thesis.
5458qed
5459
5460AOT_theorem "oa-contingent:3": ‹A!x ≡ ¬O!x›
5461 by (AOT_subst "«A!x»" "«¬¬A!x»")
5462 (auto simp add: "oth-class-taut:3:b" "oa-contingent:2"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric])
5463
5464AOT_theorem "oa-contingent:4": ‹Contingent(O!)›
5465proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:1", THEN "≡E"(2)]; rule "&I")
5466 AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5467 AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5468 then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5469 AOT_hence ‹[λx ◇E!x]a›
5470 by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5471 AOT_hence ‹O!a›
5472 by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5473 AOT_hence ‹∃x O!x› using "∃I" by blast
5474 AOT_thus ‹◇∃x O!x› using "T◇"[THEN "→E"] by blast
5475next
5476 AOT_obtain a where ‹A!a›
5477 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5478 AOT_hence ‹¬O!a› using "oa-contingent:3"[THEN "≡E"(1)] by blast
5479 AOT_hence ‹∃x ¬O!x› using "∃I" by fast
5480 AOT_thus ‹◇∃x ¬O!x› using "T◇"[THEN "→E"] by blast
5481qed
5482
5483AOT_theorem "oa-contingent:5": ‹Contingent(A!)›
5484proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:2", THEN "≡E"(2)]; rule "&I")
5485 AOT_obtain a where ‹A!a›
5486 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5487 AOT_hence ‹∃x A!x› using "∃I" by fast
5488 AOT_thus ‹◇∃x A!x› using "T◇"[THEN "→E"] by blast
5489next
5490 AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5491 AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5492 then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5493 AOT_hence ‹[λx ◇E!x]a›
5494 by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5495 AOT_hence ‹O!a›
5496 by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5497 AOT_hence ‹¬A!a› using "oa-contingent:2"[THEN "≡E"(1)] by blast
5498 AOT_hence ‹∃x ¬A!x› using "∃I" by fast
5499 AOT_thus ‹◇∃x ¬A!x› using "T◇"[THEN "→E"] by blast
5500qed
5501
5502AOT_theorem "oa-contingent:7": ‹O!⇧-x ≡ ¬A!⇧-x›
5503proof -
5504 AOT_have ‹O!x ≡ ¬A!x›
5505 using "oa-contingent:2" by blast
5506 also AOT_have ‹… ≡ A!⇧-x›
5507 using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:2"].
5508 finally AOT_have 1: ‹O!x ≡ A!⇧-x›.
5509
5510 AOT_have ‹A!x ≡ ¬O!x›
5511 using "oa-contingent:3" by blast
5512 also AOT_have ‹… ≡ O!⇧-x›
5513 using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:1"].
5514 finally AOT_have 2: ‹A!x ≡ O!⇧-x›.
5515
5516 AOT_show ‹O!⇧-x ≡ ¬A!⇧-x›
5517 using 1[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "oa-contingent:3"[of _ x] 2[symmetric]
5518 "≡E"(5) by blast
5519qed
5520
5521AOT_theorem "oa-contingent:6": ‹O!⇧- ≠ A!⇧-›
5522proof (rule "=-infix"[THEN "≡⇩d⇩fI"]; rule "raa-cor:2")
5523 AOT_assume 1: ‹O!⇧- = A!⇧-›
5524 fix x
5525 AOT_have ‹A!⇧-x ≡ O!⇧-x›
5526 apply (rule "rule=E"[rotated, OF 1]) by (fact "oth-class-taut:3:a")
5527 AOT_hence ‹A!⇧-x ≡ ¬A!⇧-x›
5528 using "oa-contingent:7" "≡E" by fast
5529 AOT_thus ‹(A!⇧-x ≡ ¬A!⇧-x) & ¬(A!⇧-x ≡ ¬A!⇧-x)› using "oth-class-taut:3:c" "&I" by blast
5530qed
5531
5532AOT_theorem "oa-contingent:8": ‹Contingent(O!⇧-)›
5533 using "thm-cont-prop:3"[unvarify F, OF "oa-exist:1", THEN "≡E"(1), OF "oa-contingent:4"].
5534
5535AOT_theorem "oa-contingent:9": ‹Contingent(A!⇧-)›
5536 using "thm-cont-prop:3"[unvarify F, OF "oa-exist:2", THEN "≡E"(1), OF "oa-contingent:5"].
5537
5538AOT_define WeaklyContingent :: ‹Π ⇒ φ› ("WeaklyContingent'(_')")
5539 "df-cont-nec": "WeaklyContingent([F]) ≡⇩d⇩f Contingent([F]) & ∀x (◇[F]x → □[F]x)"
5540
5541AOT_theorem "cont-nec-fact1:1": ‹WeaklyContingent([F]) ≡ WeaklyContingent([F]⇧-)›
5542proof -
5543 AOT_have ‹WeaklyContingent([F]) ≡ Contingent([F]) & ∀x (◇[F]x → □[F]x)›
5544 using "df-cont-nec"[THEN "≡Df"] by blast
5545 also AOT_have ‹... ≡ Contingent([F]⇧-) & ∀x (◇[F]x → □[F]x)›
5546 apply (rule "oth-class-taut:8:f"[THEN "≡E"(2)]; rule "→I")
5547 using "thm-cont-prop:3".
5548 also AOT_have ‹… ≡ Contingent([F]⇧-) & ∀x (◇[F]⇧-x → □[F]⇧-x)›
5549 proof (rule "oth-class-taut:8:e"[THEN "≡E"(2)]; rule "→I"; rule "≡I"; rule "→I"; rule GEN; rule "→I")
5550 fix x
5551 AOT_assume 0: ‹∀x (◇[F]x → □[F]x)›
5552 AOT_assume 1: ‹◇[F]⇧-x›
5553 AOT_have ‹◇¬[F]x›
5554 by (AOT_subst_rev "«[F]⇧-x»" "«¬[F]x»")
5555 (auto simp add: "thm-relation-negation:1" 1)
5556 AOT_hence 2: ‹¬□[F]x›
5557 using "KBasic:11"[THEN "≡E"(2)] by blast
5558 AOT_show ‹□[F]⇧-x›
5559 proof (rule "raa-cor:1")
5560 AOT_assume 3: ‹¬□[F]⇧-x›
5561 AOT_have ‹¬□¬[F]x›
5562 by (AOT_subst_rev "«[F]⇧-x»" "«¬[F]x»")
5563 (auto simp add: "thm-relation-negation:1" 3)
5564 AOT_hence ‹◇[F]x›
5565 using "conventions:5"[THEN "≡⇩d⇩fI"] by simp
5566 AOT_hence ‹□[F]x› using 0 "∀E" "→E" by fast
5567 AOT_thus ‹□[F]x & ¬□[F]x› using "&I" 2 by blast
5568 qed
5569 next
5570 fix x
5571 AOT_assume 0: ‹∀x (◇[F]⇧-x → □[F]⇧-x)›
5572 AOT_assume 1: ‹◇[F]x›
5573 AOT_have ‹◇¬[F]⇧-x›
5574 by (AOT_subst "«¬[F]⇧-x»" "«[F]x»")
5575 (auto simp: "thm-relation-negation:2" 1)
5576 AOT_hence 2: ‹¬□[F]⇧-x›
5577 using "KBasic:11"[THEN "≡E"(2)] by blast
5578 AOT_show ‹□[F]x›
5579 proof (rule "raa-cor:1")
5580 AOT_assume 3: ‹¬□[F]x›
5581 AOT_have ‹¬□¬[F]⇧-x›
5582 by (AOT_subst "«¬[F]⇧-x»" "«[F]x»")
5583 (auto simp add: "thm-relation-negation:2" 3)
5584 AOT_hence ‹◇[F]⇧-x›
5585 using "conventions:5"[THEN "≡⇩d⇩fI"] by simp
5586 AOT_hence ‹□[F]⇧-x› using 0 "∀E" "→E" by fast
5587 AOT_thus ‹□[F]⇧-x & ¬□[F]⇧-x› using "&I" 2 by blast
5588 qed
5589 qed
5590 also AOT_have ‹… ≡ WeaklyContingent([F]⇧-)›
5591 using "df-cont-nec"[THEN "≡Df", symmetric] by blast
5592 finally show ?thesis.
5593qed
5594
5595AOT_theorem "cont-nec-fact1:2": ‹(WeaklyContingent([F]) & ¬WeaklyContingent([G])) → F ≠ G›
5596proof (rule "→I"; rule "=-infix"[THEN "≡⇩d⇩fI"]; rule "raa-cor:2")
5597 AOT_assume 1: ‹WeaklyContingent([F]) & ¬WeaklyContingent([G])›
5598 AOT_hence ‹WeaklyContingent([F])› using "&E" by blast
5599 moreover AOT_assume ‹F = G›
5600 ultimately AOT_have ‹WeaklyContingent([G])›
5601 using "rule=E" by blast
5602 AOT_thus ‹WeaklyContingent([G]) & ¬WeaklyContingent([G])›
5603 using 1 "&I" "&E" by blast
5604qed
5605
5606AOT_theorem "cont-nec-fact2:1": ‹WeaklyContingent(O!)›
5607proof (rule "df-cont-nec"[THEN "≡⇩d⇩fI"]; rule "&I")
5608 AOT_show ‹Contingent(O!)›
5609 using "oa-contingent:4".
5610next
5611 AOT_show ‹∀x (◇[O!]x → □[O!]x)›
5612 apply (rule GEN; rule "→I")
5613 using "oa-facts:5"[THEN "≡E"(1)] by blast
5614qed
5615
5616
5617AOT_theorem "cont-nec-fact2:2": ‹WeaklyContingent(A!)›
5618proof (rule "df-cont-nec"[THEN "≡⇩d⇩fI"]; rule "&I")
5619 AOT_show ‹Contingent(A!)›
5620 using "oa-contingent:5".
5621next
5622 AOT_show ‹∀x (◇[A!]x → □[A!]x)›
5623 apply (rule GEN; rule "→I")
5624 using "oa-facts:6"[THEN "≡E"(1)] by blast
5625qed
5626
5627AOT_theorem "cont-nec-fact2:3": ‹¬WeaklyContingent(E!)›
5628proof (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5629 rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "raa-cor:2")
5630 AOT_have ‹◇∃x (E!x & ¬❙𝒜E!x)› using "qml:4"[axiom_inst].
5631 AOT_hence ‹∃x ◇(E!x & ¬❙𝒜E!x)› using "BF◇"[THEN "→E"] by blast
5632 then AOT_obtain a where ‹◇(E!a & ¬❙𝒜E!a)› using "∃E"[rotated] by blast
5633 AOT_hence 1: ‹◇E!a & ◇¬❙𝒜E!a› using "KBasic2:3"[THEN "→E"] by simp
5634 moreover AOT_assume ‹∀x (◇[E!]x → □[E!]x)›
5635 ultimately AOT_have ‹□E!a› using "&E" "∀E" "→E" by fast
5636 AOT_hence ‹❙𝒜E!a› using "nec-imp-act"[THEN "→E"] by blast
5637 AOT_hence ‹□❙𝒜E!a› using "qml-act:1"[axiom_inst, THEN "→E"] by blast
5638 moreover AOT_have ‹¬□❙𝒜E!a› using "KBasic:11"[THEN "≡E"(2)] 1[THEN "&E"(2)] by meson
5639 ultimately AOT_have ‹□❙𝒜E!a & ¬□❙𝒜E!a› using "&I" by blast
5640 AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
5641qed
5642
5643AOT_theorem "cont-nec-fact2:4": ‹¬WeaklyContingent(L)›
5644 apply (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5645 rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(1))
5646 apply (rule "contingent-properties:4"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
5647 apply (rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "useful-tautologies:2"[THEN "→E"])
5648 using "thm-noncont-e-e:3"[THEN "contingent-properties:3"[THEN "≡⇩d⇩fE"]].
5649
5650
5651AOT_theorem "cont-nec-fact2:5": ‹O! ≠ E! & O! ≠ E!⇧- & O! ≠ L & O! ≠ L⇧-›
5652proof -
5653 AOT_have 1: ‹L↓›
5654 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5655 {
5656 fix φ and Π and Π'
5657 AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if ‹φ{Π}› and ‹¬φ{Π'}›
5658 proof (rule "raa-cor:2")
5659 AOT_assume ‹φ{Π'} ≡ φ{Π}›
5660 AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5661 AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5662 qed
5663 AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5664 using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5665 } note 0 = this
5666 show ?thesis
5667 apply(safe intro!: "&I"; rule 0)
5668 using "cqt:2[concrete]"[axiom_inst] apply blast
5669 using "oa-exist:1" apply blast
5670 using "cont-nec-fact2:3" apply fast
5671 apply (rule "useful-tautologies:2"[THEN "→E"])
5672 using "cont-nec-fact2:1" apply fast
5673 using "rel-neg-T:3" apply fast
5674 using "oa-exist:1" apply blast
5675 using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5676 apply (rule "useful-tautologies:2"[THEN "→E"])
5677 using "cont-nec-fact2:1" apply blast
5678 apply (rule "=⇩d⇩fI"(2)[OF L_def]; "cqt:2[lambda]")
5679 using "oa-exist:1" apply fast
5680 using "cont-nec-fact2:4" apply fast
5681 apply (rule "useful-tautologies:2"[THEN "→E"])
5682 using "cont-nec-fact2:1" apply fast
5683 using "rel-neg-T:3" apply fast
5684 using "oa-exist:1" apply fast
5685 apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5686 apply (rule "=⇩d⇩fI"(2)[OF L_def]; "cqt:2[lambda]")
5687 apply (rule "useful-tautologies:2"[THEN "→E"])
5688 using "cont-nec-fact2:1" by blast
5689qed
5690
5691
5692AOT_theorem "cont-nec-fact2:6": ‹A! ≠ E! & A! ≠ E!⇧- & A! ≠ L & A! ≠ L⇧-›
5693proof -
5694 AOT_have 1: ‹L↓›
5695 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5696 {
5697 fix φ and Π and Π'
5698 AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if ‹φ{Π}› and ‹¬φ{Π'}›
5699 proof (rule "raa-cor:2")
5700 AOT_assume ‹φ{Π'} ≡ φ{Π}›
5701 AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5702 AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5703 qed
5704 AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5705 using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5706 } note 0 = this
5707 show ?thesis
5708 apply(safe intro!: "&I"; rule 0)
5709 using "cqt:2[concrete]"[axiom_inst] apply blast
5710 using "oa-exist:2" apply blast
5711 using "cont-nec-fact2:3" apply fast
5712 apply (rule "useful-tautologies:2"[THEN "→E"])
5713 using "cont-nec-fact2:2" apply fast
5714 using "rel-neg-T:3" apply fast
5715 using "oa-exist:2" apply blast
5716 using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5717 apply (rule "useful-tautologies:2"[THEN "→E"])
5718 using "cont-nec-fact2:2" apply blast
5719 apply (rule "=⇩d⇩fI"(2)[OF L_def]; "cqt:2[lambda]")
5720 using "oa-exist:2" apply fast
5721 using "cont-nec-fact2:4" apply fast
5722 apply (rule "useful-tautologies:2"[THEN "→E"])
5723 using "cont-nec-fact2:2" apply fast
5724 using "rel-neg-T:3" apply fast
5725 using "oa-exist:2" apply fast
5726 apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5727 apply (rule "=⇩d⇩fI"(2)[OF L_def]; "cqt:2[lambda]")
5728 apply (rule "useful-tautologies:2"[THEN "→E"])
5729 using "cont-nec-fact2:2" by blast
5730qed
5731
5732AOT_define necessary_or_contingently_false :: ‹φ ⇒ φ› ("❙Δ_" [49] 54)
5733 ‹❙Δp ≡⇩d⇩f □p ∨ (¬❙𝒜p & ◇p)›
5734
5735AOT_theorem sixteen:
5736 shows ‹∃F⇩1∃F⇩2∃F⇩3∃F⇩4∃F⇩5∃F⇩6∃F⇩7∃F⇩8∃F⇩9∃F⇩1⇩0∃F⇩1⇩1∃F⇩1⇩2∃F⇩1⇩3∃F⇩1⇩4∃F⇩1⇩5∃F⇩1⇩6 (
5737«F⇩1::<κ>» ≠ F⇩2 & F⇩1 ≠ F⇩3 & F⇩1 ≠ F⇩4 & F⇩1 ≠ F⇩5 & F⇩1 ≠ F⇩6 & F⇩1 ≠ F⇩7 & F⇩1 ≠ F⇩8 & F⇩1 ≠ F⇩9 & F⇩1 ≠ F⇩1⇩0 & F⇩1 ≠ F⇩1⇩1 & F⇩1 ≠ F⇩1⇩2 & F⇩1 ≠ F⇩1⇩3 & F⇩1 ≠ F⇩1⇩4 & F⇩1 ≠ F⇩1⇩5 & F⇩1 ≠ F⇩1⇩6 &
5738F⇩2 ≠ F⇩3 & F⇩2 ≠ F⇩4 & F⇩2 ≠ F⇩5 & F⇩2 ≠ F⇩6 & F⇩2 ≠ F⇩7 & F⇩2 ≠ F⇩8 & F⇩2 ≠ F⇩9 & F⇩2 ≠ F⇩1⇩0 & F⇩2 ≠ F⇩1⇩1 & F⇩2 ≠ F⇩1⇩2 & F⇩2 ≠ F⇩1⇩3 & F⇩2 ≠ F⇩1⇩4 & F⇩2 ≠ F⇩1⇩5 & F⇩2 ≠ F⇩1⇩6 &
5739F⇩3 ≠ F⇩4 & F⇩3 ≠ F⇩5 & F⇩3 ≠ F⇩6 & F⇩3 ≠ F⇩7 & F⇩3 ≠ F⇩8 & F⇩3 ≠ F⇩9 & F⇩3 ≠ F⇩1⇩0 & F⇩3 ≠ F⇩1⇩1 & F⇩3 ≠ F⇩1⇩2 & F⇩3 ≠ F⇩1⇩3 & F⇩3 ≠ F⇩1⇩4 & F⇩3 ≠ F⇩1⇩5 & F⇩3 ≠ F⇩1⇩6 &
5740F⇩4 ≠ F⇩5 & F⇩4 ≠ F⇩6 & F⇩4 ≠ F⇩7 & F⇩4 ≠ F⇩8 & F⇩4 ≠ F⇩9 & F⇩4 ≠ F⇩1⇩0 & F⇩4 ≠ F⇩1⇩1 & F⇩4 ≠ F⇩1⇩2 & F⇩4 ≠ F⇩1⇩3 & F⇩4 ≠ F⇩1⇩4 & F⇩4 ≠ F⇩1⇩5 & F⇩4 ≠ F⇩1⇩6 &
5741F⇩5 ≠ F⇩6 & F⇩5 ≠ F⇩7 & F⇩5 ≠ F⇩8 & F⇩5 ≠ F⇩9 & F⇩5 ≠ F⇩1⇩0 & F⇩5 ≠ F⇩1⇩1 & F⇩5 ≠ F⇩1⇩2 & F⇩5 ≠ F⇩1⇩3 & F⇩5 ≠ F⇩1⇩4 & F⇩5 ≠ F⇩1⇩5 & F⇩5 ≠ F⇩1⇩6 &
5742F⇩6 ≠ F⇩7 & F⇩6 ≠ F⇩8 & F⇩6 ≠ F⇩9 & F⇩6 ≠ F⇩1⇩0 & F⇩6 ≠ F⇩1⇩1 & F⇩6 ≠ F⇩1⇩2 & F⇩6 ≠ F⇩1⇩3 & F⇩6 ≠ F⇩1⇩4 & F⇩6 ≠ F⇩1⇩5 & F⇩6 ≠ F⇩1⇩6 &
5743F⇩7 ≠ F⇩8 & F⇩7 ≠ F⇩9 & F⇩7 ≠ F⇩1⇩0 & F⇩7 ≠ F⇩1⇩1 & F⇩7 ≠ F⇩1⇩2 & F⇩7 ≠ F⇩1⇩3 & F⇩7 ≠ F⇩1⇩4 & F⇩7 ≠ F⇩1⇩5 & F⇩7 ≠ F⇩1⇩6 &
5744F⇩8 ≠ F⇩9 & F⇩8 ≠ F⇩1⇩0 & F⇩8 ≠ F⇩1⇩1 & F⇩8 ≠ F⇩1⇩2 & F⇩8 ≠ F⇩1⇩3 & F⇩8 ≠ F⇩1⇩4 & F⇩8 ≠ F⇩1⇩5 & F⇩8 ≠ F⇩1⇩6 &
5745F⇩9 ≠ F⇩1⇩0 & F⇩9 ≠ F⇩1⇩1 & F⇩9 ≠ F⇩1⇩2 & F⇩9 ≠ F⇩1⇩3 & F⇩9 ≠ F⇩1⇩4 & F⇩9 ≠ F⇩1⇩5 & F⇩9 ≠ F⇩1⇩6 &
5746F⇩1⇩0 ≠ F⇩1⇩1 & F⇩1⇩0 ≠ F⇩1⇩2 & F⇩1⇩0 ≠ F⇩1⇩3 & F⇩1⇩0 ≠ F⇩1⇩4 & F⇩1⇩0 ≠ F⇩1⇩5 & F⇩1⇩0 ≠ F⇩1⇩6 &
5747F⇩1⇩1 ≠ F⇩1⇩2 & F⇩1⇩1 ≠ F⇩1⇩3 & F⇩1⇩1 ≠ F⇩1⇩4 & F⇩1⇩1 ≠ F⇩1⇩5 & F⇩1⇩1 ≠ F⇩1⇩6 &
5748F⇩1⇩2 ≠ F⇩1⇩3 & F⇩1⇩2 ≠ F⇩1⇩4 & F⇩1⇩2 ≠ F⇩1⇩5 & F⇩1⇩2 ≠ F⇩1⇩6 &
5749F⇩1⇩3 ≠ F⇩1⇩4 & F⇩1⇩3 ≠ F⇩1⇩5 & F⇩1⇩3 ≠ F⇩1⇩6 &
5750F⇩1⇩4 ≠ F⇩1⇩5 & F⇩1⇩4 ≠ F⇩1⇩6 &
5751F⇩1⇩5 ≠ F⇩1⇩6)›
5752proof -
5753
5754 AOT_have Delta_pos: ‹❙Δφ → ◇φ› for φ
5755 proof(rule "→I")
5756 AOT_assume ‹❙Δφ›
5757 AOT_hence ‹□φ ∨ (¬❙𝒜φ & ◇φ)›
5758 using "≡⇩d⇩fE"[OF necessary_or_contingently_false] by blast
5759 moreover {
5760 AOT_assume ‹□φ›
5761 AOT_hence ‹◇φ›
5762 by (metis "B◇" "T◇" "vdash-properties:10")
5763 }
5764 moreover {
5765 AOT_assume ‹¬❙𝒜φ & ◇φ›
5766 AOT_hence ‹◇φ›
5767 using "&E" by blast
5768 }
5769 ultimately AOT_show ‹◇φ›
5770 by (metis "∨E"(2) "raa-cor:1")
5771 qed
5772
5773 AOT_have act_and_not_nec_not_delta: ‹¬❙Δφ› if ‹❙𝒜φ› and ‹¬□φ› for φ
5774 using "≡⇩d⇩fE" "&E"(1) "∨E"(2) necessary_or_contingently_false "raa-cor:3" that(1) that(2) by blast
5775 AOT_have act_and_pos_not_not_delta: ‹¬❙Δφ› if ‹❙𝒜φ› and ‹◇¬φ› for φ
5776 using "KBasic:11" act_and_not_nec_not_delta "≡E"(2) that(1) that(2) by blast
5777 AOT_have impossible_delta: ‹¬❙Δφ› if ‹¬◇φ› for φ
5778 using Delta_pos "modus-tollens:1" that by blast
5779 AOT_have not_act_and_pos_delta: ‹❙Δφ› if ‹¬❙𝒜φ› and ‹◇φ› for φ
5780 by (meson "≡⇩d⇩fI" "&I" "∨I"(2) necessary_or_contingently_false that(1) that(2))
5781 AOT_have nec_delta: ‹❙Δφ› if ‹□φ› for φ
5782 using "≡⇩d⇩fI" "∨I"(1) necessary_or_contingently_false that by blast
5783
5784 AOT_obtain a where a_prop: ‹A!a›
5785 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5786 AOT_obtain b where b_prop: ‹◇[E!]b & ¬❙𝒜[E!]b›
5787 using "pos-not-pna:3" using "∃E"[rotated] by blast
5788
5789 AOT_have b_ord: ‹[O!]b›
5790 proof(rule "=⇩d⇩fI"(2)[OF AOT_ordinary])
5791 AOT_show ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
5792 next
5793 AOT_show ‹[λx ◇[E!]x]b›
5794 proof (rule "β←C"(1); ("cqt:2[lambda]")?)
5795 AOT_show ‹b↓› by (rule "cqt:2[const_var]"[axiom_inst])
5796 AOT_show ‹◇[E!]b› by (fact b_prop[THEN "&E"(1)])
5797 qed
5798 qed
5799
5800 AOT_have nec_not_L_neg: ‹□¬[L⇧-]x› for x
5801 using "thm-noncont-e-e:2" "contingent-properties:2"[THEN "≡⇩d⇩fE"] "&E"
5802 CBF[THEN "→E"] "∀E" by blast
5803 AOT_have nec_L: ‹□[L]x› for x
5804 using "thm-noncont-e-e:1" "contingent-properties:1"[THEN "≡⇩d⇩fE"]
5805 CBF[THEN "→E"] "∀E" by blast
5806
5807 AOT_have act_ord_b: ‹❙𝒜[O!]b›
5808 using b_ord "≡E"(1) "oa-facts:7" by blast
5809 AOT_have delta_ord_b: ‹❙Δ[O!]b›
5810 by (meson "≡⇩d⇩fI" b_ord "∨I"(1) necessary_or_contingently_false "oa-facts:1" "vdash-properties:10")
5811 AOT_have not_act_ord_a: ‹¬❙𝒜[O!]a›
5812 by (meson a_prop "≡E"(1) "≡E"(3) "oa-contingent:3" "oa-facts:7")
5813 AOT_have not_delta_ord_a: ‹¬❙Δ[O!]a›
5814 by (metis Delta_pos "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "reductio-aa:1" "vdash-properties:10")
5815
5816 AOT_have not_act_abs_b: ‹¬❙𝒜[A!]b›
5817 by (meson b_ord "≡E"(1) "≡E"(3) "oa-contingent:2" "oa-facts:8")
5818 AOT_have not_delta_abs_b: ‹¬❙Δ[A!]b›
5819 proof(rule "raa-cor:2")
5820 AOT_assume ‹❙Δ[A!]b›
5821 AOT_hence ‹◇[A!]b›
5822 by (metis Delta_pos "vdash-properties:10")
5823 AOT_thus ‹[A!]b & ¬[A!]b›
5824 by (metis b_ord "&I" "≡E"(1) "oa-contingent:2" "oa-facts:4" "vdash-properties:10")
5825 qed
5826 AOT_have act_abs_a: ‹❙𝒜[A!]a›
5827 using a_prop "≡E"(1) "oa-facts:8" by blast
5828 AOT_have delta_abs_a: ‹❙Δ[A!]a›
5829 by (metis "≡⇩d⇩fI" a_prop "oa-facts:2" "vdash-properties:10" "∨I"(1) necessary_or_contingently_false)
5830
5831 AOT_have not_act_concrete_b: ‹¬❙𝒜[E!]b›
5832 using b_prop "&E"(2) by blast
5833 AOT_have delta_concrete_b: ‹❙Δ[E!]b›
5834 proof (rule "≡⇩d⇩fI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5835 AOT_show ‹¬❙𝒜[E!]b› using b_prop "&E"(2) by blast
5836 next
5837 AOT_show ‹◇[E!]b› using b_prop "&E"(1) by blast
5838 qed
5839 AOT_have not_act_concrete_a: ‹¬❙𝒜[E!]a›
5840 proof (rule "raa-cor:2")
5841 AOT_assume ‹❙𝒜[E!]a›
5842 AOT_hence 1: ‹◇[E!]a› by (metis "Act-Sub:3" "vdash-properties:10")
5843 AOT_have ‹[A!]a› by (simp add: a_prop)
5844 AOT_hence ‹[λx ¬◇[E!]x]a›
5845 by (rule "=⇩d⇩fE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5846 AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5847 AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5848 qed
5849 AOT_have not_delta_concrete_a: ‹¬❙Δ[E!]a›
5850 proof (rule "raa-cor:2")
5851 AOT_assume ‹❙Δ[E!]a›
5852 AOT_hence 1: ‹◇[E!]a› by (metis Delta_pos "vdash-properties:10")
5853 AOT_have ‹[A!]a› by (simp add: a_prop)
5854 AOT_hence ‹[λx ¬◇[E!]x]a›
5855 by (rule "=⇩d⇩fE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5856 AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5857 AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5858 qed
5859
5860 AOT_have not_act_q_zero: ‹¬❙𝒜q⇩0›
5861 by (meson "log-prop-prop:2" "pos-not-pna:1" q⇩0_def "reductio-aa:1" "rule-id-def:2:a[zero]")
5862 AOT_have delta_q_zero: ‹❙Δq⇩0›
5863 proof(rule "≡⇩d⇩fI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5864 AOT_show ‹¬❙𝒜q⇩0› using not_act_q_zero.
5865 AOT_show ‹◇q⇩0› by (meson "&E"(1) q⇩0_prop)
5866 qed
5867 AOT_have act_not_q_zero: ‹❙𝒜¬q⇩0› using "Act-Basic:1" "∨E"(2) not_act_q_zero by blast
5868 AOT_have not_delta_not_q_zero: ‹¬❙Δ¬q⇩0›
5869 using "≡⇩d⇩fE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta "&E"(1) "∨E"(2) not_act_q_zero q⇩0_prop by blast
5870
5871 AOT_have ‹[L⇧-]↓› by (simp add: "rel-neg-T:3")
5872 moreover AOT_have ‹¬❙𝒜[L⇧-]b & ¬❙Δ[L⇧-]b & ¬❙𝒜[L⇧-]a & ¬❙Δ[L⇧-]a›
5873 proof (safe intro!: "&I")
5874 AOT_show ‹¬❙𝒜[L⇧-]b› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5875 AOT_show ‹¬❙Δ[L⇧-]b› by (meson Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg)
5876 AOT_show ‹¬❙𝒜[L⇧-]a› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5877 AOT_show ‹¬❙Δ[L⇧-]a› using Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg by blast
5878 qed
5879 ultimately AOT_obtain F⇩0 where ‹¬❙𝒜[F⇩0]b & ¬❙Δ[F⇩0]b & ¬❙𝒜[F⇩0]a & ¬❙Δ[F⇩0]a›
5880 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5881 AOT_hence ‹¬❙𝒜[F⇩0]b› and ‹¬❙Δ[F⇩0]b› and ‹¬❙𝒜[F⇩0]a› and ‹¬❙Δ[F⇩0]a›
5882 using "&E" by blast+
5883 note props = this
5884
5885 let ?Π = "«[λy [A!]y & q⇩0]»"
5886 AOT_modally_strict {
5887 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5888 } note 1 = this
5889 moreover AOT_have‹¬❙𝒜[«?Π»]b & ¬❙Δ[«?Π»]b & ¬❙𝒜[«?Π»]a & ❙Δ[«?Π»]a›
5890 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5891 AOT_show ‹¬❙𝒜([A!]b & q⇩0)›
5892 using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5893 next AOT_show ‹¬❙Δ([A!]b & q⇩0)›
5894 by (metis Delta_pos "KBasic2:3" "&E"(1) "≡E"(4) not_act_abs_b "oa-facts:4" "oa-facts:8" "raa-cor:3" "vdash-properties:10")
5895 next AOT_show ‹¬❙𝒜([A!]a & q⇩0)›
5896 using "Act-Basic:2" "&E"(2) "≡E"(1) not_act_q_zero "raa-cor:3" by blast
5897 next AOT_show ‹❙Δ([A!]a & q⇩0)›
5898 proof (rule not_act_and_pos_delta)
5899 AOT_show ‹¬❙𝒜([A!]a & q⇩0)›
5900 using "Act-Basic:2" "&E"(2) "≡E"(4) not_act_q_zero "raa-cor:3" by blast
5901 next AOT_show ‹◇([A!]a & q⇩0)›
5902 by (metis "&I" "→E" Delta_pos "KBasic:16" "&E"(1) delta_abs_a "≡E"(1) "oa-facts:6" q⇩0_prop)
5903 qed
5904 qed
5905 ultimately AOT_obtain F⇩1 where ‹¬❙𝒜[F⇩1]b & ¬❙Δ[F⇩1]b & ¬❙𝒜[F⇩1]a & ❙Δ[F⇩1]a›
5906 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5907 AOT_hence ‹¬❙𝒜[F⇩1]b› and ‹¬❙Δ[F⇩1]b› and ‹¬❙𝒜[F⇩1]a› and ‹❙Δ[F⇩1]a›
5908 using "&E" by blast+
5909 note props = props this
5910
5911 let ?Π = "«[λy [A!]y & ¬q⇩0]»"
5912 AOT_modally_strict {
5913 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5914 } note 1 = this
5915 moreover AOT_have ‹¬❙𝒜[«?Π»]b & ¬❙Δ[«?Π»]b & ❙𝒜[«?Π»]a & ¬❙Δ[«?Π»]a›
5916 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5917 AOT_show ‹¬❙𝒜([A!]b & ¬q⇩0)›
5918 using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5919 next AOT_show ‹¬❙Δ([A!]b & ¬q⇩0)›
5920 by (meson "RM◇" Delta_pos "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b "oa-facts:4" "oa-facts:8")
5921 next AOT_show ‹❙𝒜([A!]a & ¬q⇩0)›
5922 by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
5923 next AOT_show ‹¬❙Δ([A!]a & ¬q⇩0)›
5924 proof (rule act_and_not_nec_not_delta)
5925 AOT_show ‹❙𝒜([A!]a & ¬q⇩0)›
5926 by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
5927 next
5928 AOT_show ‹¬□([A!]a & ¬q⇩0)›
5929 by (metis "KBasic2:1" "KBasic:3" "&E"(1) "&E"(2) "≡E"(4) q⇩0_prop "raa-cor:3")
5930 qed
5931 qed
5932 ultimately AOT_obtain F⇩2 where ‹¬❙𝒜[F⇩2]b & ¬❙Δ[F⇩2]b & ❙𝒜[F⇩2]a & ¬❙Δ[F⇩2]a›
5933 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5934 AOT_hence ‹¬❙𝒜[F⇩2]b› and ‹¬❙Δ[F⇩2]b› and ‹❙𝒜[F⇩2]a› and ‹¬❙Δ[F⇩2]a›
5935 using "&E" by blast+
5936 note props = props this
5937
5938 AOT_have abstract_prop: ‹¬❙𝒜[A!]b & ¬❙Δ[A!]b & ❙𝒜[A!]a & ❙Δ[A!]a›
5939 using act_abs_a "&I" delta_abs_a not_act_abs_b not_delta_abs_b by presburger
5940 then AOT_obtain F⇩3 where ‹¬❙𝒜[F⇩3]b & ¬❙Δ[F⇩3]b & ❙𝒜[F⇩3]a & ❙Δ[F⇩3]a›
5941 using "∃I"(1)[rotated, THEN "∃E"[rotated]] "oa-exist:2" by fastforce
5942 AOT_hence ‹¬❙𝒜[F⇩3]b› and ‹¬❙Δ[F⇩3]b› and ‹❙𝒜[F⇩3]a› and ‹❙Δ[F⇩3]a›
5943 using "&E" by blast+
5944 note props = props this
5945
5946 AOT_have ‹¬❙𝒜[E!]b & ❙Δ[E!]b & ¬❙𝒜[E!]a & ¬❙Δ[E!]a›
5947 by (meson "&I" delta_concrete_b not_act_concrete_a not_act_concrete_b not_delta_concrete_a)
5948 then AOT_obtain F⇩4 where ‹¬❙𝒜[F⇩4]b & ❙Δ[F⇩4]b & ¬❙𝒜[F⇩4]a & ¬❙Δ[F⇩4]a›
5949 using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5950 AOT_hence ‹¬❙𝒜[F⇩4]b› and ‹❙Δ[F⇩4]b› and ‹¬❙𝒜[F⇩4]a› and ‹¬❙Δ[F⇩4]a›
5951 using "&E" by blast+
5952 note props = props this
5953
5954 AOT_modally_strict {
5955 AOT_have ‹[λy q⇩0]↓› by "cqt:2[lambda]"
5956 } note 1 = this
5957 moreover AOT_have ‹¬❙𝒜[λy q⇩0]b & ❙Δ[λy q⇩0]b & ¬❙𝒜[λy q⇩0]a & ❙Δ[λy q⇩0]a›
5958 by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5959 (auto simp: not_act_q_zero delta_q_zero)
5960 ultimately AOT_obtain F⇩5 where ‹¬❙𝒜[F⇩5]b & ❙Δ[F⇩5]b & ¬❙𝒜[F⇩5]a & ❙Δ[F⇩5]a›
5961 using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5962 AOT_hence ‹¬❙𝒜[F⇩5]b› and ‹❙Δ[F⇩5]b› and ‹¬❙𝒜[F⇩5]a› and ‹❙Δ[F⇩5]a›
5963 using "&E" by blast+
5964 note props = props this
5965
5966 let ?Π = "«[λy [E!]y ∨ ([A!]y & ¬q⇩0)]»"
5967 AOT_modally_strict {
5968 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5969 } note 1 = this
5970 moreover AOT_have ‹¬❙𝒜[«?Π»]b & ❙Δ[«?Π»]b & ❙𝒜[«?Π»]a & ¬❙Δ[«?Π»]a›
5971 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5972 AOT_have ‹❙𝒜¬([A!]b & ¬q⇩0)›
5973 by (metis "Act-Basic:1" "Act-Basic:2" abstract_prop "&E"(1) "∨E"(2)
5974 "≡E"(1) "raa-cor:3")
5975 moreover AOT_have ‹¬❙𝒜[E!]b›
5976 using b_prop "&E"(2) by blast
5977 ultimately AOT_have 2: ‹❙𝒜(¬[E!]b & ¬([A!]b & ¬q⇩0))›
5978 by (metis "Act-Basic:2" "Act-Sub:1" "&I" "≡E"(3) "raa-cor:1")
5979 AOT_have ‹❙𝒜¬([E!]b ∨ ([A!]b & ¬q⇩0))›
5980 by (AOT_subst ‹«¬([E!]b ∨ ([A!]b & ¬q⇩0))»› ‹«¬[E!]b & ¬([A!]b & ¬q⇩0)»›)
5981 (auto simp: "oth-class-taut:5:d" 2)
5982 AOT_thus ‹¬❙𝒜([E!]b ∨ ([A!]b & ¬q⇩0))›
5983 by (metis "¬¬I" "Act-Sub:1" "≡E"(4))
5984 next
5985 AOT_show ‹❙Δ([E!]b ∨ ([A!]b & ¬q⇩0))›
5986 proof (rule not_act_and_pos_delta)
5987 AOT_show ‹¬❙𝒜([E!]b ∨ ([A!]b & ¬q⇩0))›
5988 by (metis "Act-Basic:2" "Act-Basic:9" "∨E"(2) "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b not_act_concrete_b "raa-cor:3")
5989 next
5990 AOT_show ‹◇([E!]b ∨ ([A!]b & ¬q⇩0))›
5991 using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
5992 qed
5993 next AOT_show ‹❙𝒜([E!]a ∨ ([A!]a & ¬q⇩0))›
5994 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
5995 next AOT_show ‹¬❙Δ([E!]a ∨ ([A!]a & ¬q⇩0))›
5996 proof (rule act_and_not_nec_not_delta)
5997 AOT_show ‹❙𝒜([E!]a ∨ ([A!]a & ¬q⇩0))›
5998 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
5999 next
6000 AOT_have ‹□¬[E!]a›
6001 by (metis "≡⇩d⇩fI" "conventions:5" "&I" "∨I"(2) necessary_or_contingently_false not_act_concrete_a not_delta_concrete_a "raa-cor:3")
6002 moreover AOT_have ‹◇¬([A!]a & ¬q⇩0)›
6003 by (metis "KBasic2:1" "KBasic:11" "KBasic:3" "&E"(1) "&E"(2) "≡E"(1) q⇩0_prop "raa-cor:3")
6004 ultimately AOT_have ‹◇(¬[E!]a & ¬([A!]a & ¬q⇩0))› by (metis "KBasic:16" "&I" "vdash-properties:10")
6005 AOT_hence ‹◇¬([E!]a ∨ ([A!]a & ¬q⇩0))›
6006 by (metis "RE◇" "≡E"(2) "oth-class-taut:5:d")
6007 AOT_thus ‹¬□([E!]a ∨ ([A!]a & ¬q⇩0))› by (metis "KBasic:12" "≡E"(1) "raa-cor:3")
6008 qed
6009 qed
6010 ultimately AOT_obtain F⇩6 where ‹¬❙𝒜[F⇩6]b & ❙Δ[F⇩6]b & ❙𝒜[F⇩6]a & ¬❙Δ[F⇩6]a›
6011 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6012 AOT_hence ‹¬❙𝒜[F⇩6]b› and ‹❙Δ[F⇩6]b› and ‹❙𝒜[F⇩6]a› and ‹¬❙Δ[F⇩6]a›
6013 using "&E" by blast+
6014 note props = props this
6015
6016 let ?Π = "«[λy [A!]y ∨ [E!]y]»"
6017 AOT_modally_strict {
6018 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6019 } note 1 = this
6020 moreover AOT_have ‹¬❙𝒜[«?Π»]b & ❙Δ[«?Π»]b & ❙𝒜[«?Π»]a & ❙Δ[«?Π»]a›
6021 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6022 AOT_show ‹¬❙𝒜([A!]b ∨ [E!]b)›
6023 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6024 next AOT_show ‹❙Δ([A!]b ∨ [E!]b)›
6025 proof (rule not_act_and_pos_delta)
6026 AOT_show ‹¬❙𝒜([A!]b ∨ [E!]b)›
6027 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6028 next AOT_show ‹◇([A!]b ∨ [E!]b)›
6029 using "KBasic2:2" b_prop "&E"(1) "∨I"(2) "≡E"(2) by blast
6030 qed
6031 next AOT_show ‹❙𝒜([A!]a ∨ [E!]a)›
6032 by (meson "Act-Basic:9" act_abs_a "∨I"(1) "≡E"(2))
6033 next AOT_show ‹❙Δ([A!]a ∨ [E!]a) ›
6034 proof (rule nec_delta)
6035 AOT_show ‹□([A!]a ∨ [E!]a)›
6036 by (metis "KBasic:15" act_abs_a act_and_not_nec_not_delta "Disjunction Addition"(1) delta_abs_a "raa-cor:3" "vdash-properties:10")
6037 qed
6038 qed
6039 ultimately AOT_obtain F⇩7 where ‹¬❙𝒜[F⇩7]b & ❙Δ[F⇩7]b & ❙𝒜[F⇩7]a & ❙Δ[F⇩7]a›
6040 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6041 AOT_hence ‹¬❙𝒜[F⇩7]b› and ‹❙Δ[F⇩7]b› and ‹❙𝒜[F⇩7]a› and ‹❙Δ[F⇩7]a›
6042 using "&E" by blast+
6043 note props = props this
6044
6045 let ?Π = "«[λy [O!]y & ¬[E!]y]»"
6046 AOT_modally_strict {
6047 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6048 } note 1 = this
6049 moreover AOT_have ‹❙𝒜[«?Π»]b & ¬❙Δ[«?Π»]b & ¬❙𝒜[«?Π»]a & ¬❙Δ[«?Π»]a›
6050 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6051 AOT_show ‹❙𝒜([O!]b & ¬[E!]b)›
6052 by (metis "Act-Basic:1" "Act-Basic:2" act_ord_b "&I" "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:3")
6053 next AOT_show ‹¬❙Δ([O!]b & ¬[E!]b)›
6054 by (metis (no_types, hide_lams) "conventions:5" "Act-Sub:1" "RM:1" act_and_not_nec_not_delta "act-conj-act:3"
6055 act_ord_b b_prop "&I" "&E"(1) "Conjunction Simplification"(2) "df-rules-formulas[3]"
6056 "≡E"(3) "raa-cor:1" "→E")
6057 next AOT_show ‹¬❙𝒜([O!]a & ¬[E!]a)›
6058 using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_ord_a "raa-cor:3" by blast
6059 next AOT_have ‹¬◇([O!]a & ¬[E!]a)›
6060 by (metis "KBasic2:3" "&E"(1) "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "raa-cor:3" "vdash-properties:10")
6061 AOT_thus ‹¬❙Δ([O!]a & ¬[E!]a)›
6062 by (rule impossible_delta)
6063 qed
6064 ultimately AOT_obtain F⇩8 where ‹❙𝒜[F⇩8]b & ¬❙Δ[F⇩8]b & ¬❙𝒜[F⇩8]a & ¬❙Δ[F⇩8]a›
6065 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6066 AOT_hence ‹❙𝒜[F⇩8]b› and ‹¬❙Δ[F⇩8]b› and ‹¬❙𝒜[F⇩8]a› and ‹¬❙Δ[F⇩8]a›
6067 using "&E" by blast+
6068 note props = props this
6069
6070
6071 let ?Π = "«[λy ¬[E!]y & ([O!]y ∨ q⇩0)]»"
6072 AOT_modally_strict {
6073 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6074 } note 1 = this
6075 moreover AOT_have ‹❙𝒜[«?Π»]b & ¬❙Δ[«?Π»]b & ¬❙𝒜[«?Π»]a & ❙Δ[«?Π»]a›
6076 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6077 AOT_show ‹❙𝒜(¬[E!]b & ([O!]b ∨ q⇩0))›
6078 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6079 "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6080 next AOT_show ‹¬❙Δ(¬[E!]b & ([O!]b ∨ q⇩0))›
6081 proof (rule act_and_pos_not_not_delta)
6082 AOT_show ‹❙𝒜(¬[E!]b & ([O!]b ∨ q⇩0))›
6083 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6084 "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6085 next
6086 AOT_show ‹◇¬(¬[E!]b & ([O!]b ∨ q⇩0))›
6087 proof (AOT_subst ‹«¬(¬[E!]b & ([O!]b ∨ q⇩0))»› ‹«[E!]b ∨ ¬([O!]b ∨ q⇩0)»›)
6088 AOT_modally_strict {
6089 AOT_show ‹¬(¬[E!]b & ([O!]b ∨ q⇩0)) ≡ [E!]b ∨ ¬([O!]b ∨ q⇩0)›
6090 by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2) "∨E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
6091 }
6092 next
6093 AOT_show ‹◇([E!]b ∨ ¬([O!]b ∨ q⇩0))›
6094 using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6095 qed
6096 qed
6097 next
6098 AOT_show ‹¬❙𝒜(¬[E!]a & ([O!]a ∨ q⇩0))›
6099 using "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2" by blast
6100 next
6101 AOT_show ‹❙Δ(¬[E!]a & ([O!]a ∨ q⇩0))›
6102 proof (rule not_act_and_pos_delta)
6103 AOT_show ‹¬❙𝒜(¬[E!]a & ([O!]a ∨ q⇩0))›
6104 by (metis "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2")
6105 next
6106 AOT_have ‹□¬[E!]a›
6107 using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "raa-cor:5" by blast
6108 moreover AOT_have ‹◇([O!]a ∨ q⇩0)›
6109 by (metis "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(3) q⇩0_prop "raa-cor:3")
6110 ultimately AOT_show ‹◇(¬[E!]a & ([O!]a ∨ q⇩0))›
6111 by (metis "KBasic:16" "&I" "vdash-properties:10")
6112 qed
6113 qed
6114 ultimately AOT_obtain F⇩9 where ‹❙𝒜[F⇩9]b & ¬❙Δ[F⇩9]b & ¬❙𝒜[F⇩9]a & ❙Δ[F⇩9]a›
6115 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6116 AOT_hence ‹❙𝒜[F⇩9]b› and ‹¬❙Δ[F⇩9]b› and ‹¬❙𝒜[F⇩9]a› and ‹❙Δ[F⇩9]a›
6117 using "&E" by blast+
6118 note props = props this
6119
6120 AOT_modally_strict {
6121 AOT_have ‹[λy ¬q⇩0]↓› by "cqt:2[lambda]"
6122 } note 1 = this
6123 moreover AOT_have ‹❙𝒜[λy ¬q⇩0]b & ¬❙Δ[λy ¬q⇩0]b & ❙𝒜[λy ¬q⇩0]a & ¬❙Δ[λy ¬q⇩0]a›
6124 by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1]; auto simp: act_not_q_zero not_delta_not_q_zero)
6125 ultimately AOT_obtain F⇩1⇩0 where ‹❙𝒜[F⇩1⇩0]b & ¬❙Δ[F⇩1⇩0]b & ❙𝒜[F⇩1⇩0]a & ¬❙Δ[F⇩1⇩0]a›
6126 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6127 AOT_hence ‹❙𝒜[F⇩1⇩0]b› and ‹¬❙Δ[F⇩1⇩0]b› and ‹❙𝒜[F⇩1⇩0]a› and ‹¬❙Δ[F⇩1⇩0]a›
6128 using "&E" by blast+
6129 note props = props this
6130
6131 AOT_modally_strict {
6132 AOT_have ‹[λy ¬[E!]y]↓› by "cqt:2[lambda]"
6133 } note 1 = this
6134 moreover AOT_have ‹❙𝒜[λy ¬[E!]y]b & ¬❙Δ[λy ¬[E!]y]b & ❙𝒜[λy ¬[E!]y]a & ❙Δ[λy ¬[E!]y]a›
6135 proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6136 AOT_show ‹❙𝒜¬[E!]b›
6137 using "Act-Basic:1" "∨E"(2) not_act_concrete_b by blast
6138 next AOT_show ‹¬❙Δ¬[E!]b›
6139 using "≡⇩d⇩fE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta b_prop "&E"(1) "∨E"(2) not_act_concrete_b by blast
6140 next AOT_show ‹❙𝒜¬[E!]a›
6141 using "Act-Basic:1" "∨E"(2) not_act_concrete_a by blast
6142 next AOT_show ‹❙Δ¬[E!]a›
6143 using "KBasic2:1" "≡E"(2) nec_delta not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "reductio-aa:1" by blast
6144 qed
6145 ultimately AOT_obtain F⇩1⇩1 where ‹❙𝒜[F⇩1⇩1]b & ¬❙Δ[F⇩1⇩1]b & ❙𝒜[F⇩1⇩1]a & ❙Δ[F⇩1⇩1]a›
6146 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6147 AOT_hence ‹❙𝒜[F⇩1⇩1]b› and ‹¬❙Δ[F⇩1⇩1]b› and ‹❙𝒜[F⇩1⇩1]a› and ‹❙Δ[F⇩1⇩1]a›
6148 using "&E" by blast+
6149 note props = props this
6150
6151 AOT_have ‹❙𝒜[O!]b & ❙Δ[O!]b & ¬❙𝒜[O!]a & ¬❙Δ[O!]a›
6152 by (simp add: act_ord_b "&I" delta_ord_b not_act_ord_a not_delta_ord_a)
6153 then AOT_obtain F⇩1⇩2 where ‹❙𝒜[F⇩1⇩2]b & ❙Δ[F⇩1⇩2]b & ¬❙𝒜[F⇩1⇩2]a & ¬❙Δ[F⇩1⇩2]a›
6154 using "oa-exist:1" "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6155 AOT_hence ‹❙𝒜[F⇩1⇩2]b› and ‹❙Δ[F⇩1⇩2]b› and ‹¬❙𝒜[F⇩1⇩2]a› and ‹¬❙Δ[F⇩1⇩2]a›
6156 using "&E" by blast+
6157 note props = props this
6158
6159 let ?Π = "«[λy [O!]y ∨ q⇩0]»"
6160 AOT_modally_strict {
6161 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6162 } note 1 = this
6163 moreover AOT_have ‹❙𝒜[«?Π»]b & ❙Δ[«?Π»]b & ¬❙𝒜[«?Π»]a & ❙Δ[«?Π»]a›
6164 proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6165 AOT_show ‹❙𝒜([O!]b ∨ q⇩0)›
6166 by (meson "Act-Basic:9" act_ord_b "∨I"(1) "≡E"(2))
6167 next AOT_show ‹❙Δ([O!]b ∨ q⇩0)›
6168 by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6169 next AOT_show ‹¬❙𝒜([O!]a ∨ q⇩0)›
6170 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6171 next AOT_show ‹❙Δ([O!]a ∨ q⇩0)›
6172 proof (rule not_act_and_pos_delta)
6173 AOT_show ‹¬❙𝒜([O!]a ∨ q⇩0)›
6174 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6175 next AOT_show ‹◇([O!]a ∨ q⇩0)›
6176 using "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(2) q⇩0_prop by blast
6177 qed
6178 qed
6179 ultimately AOT_obtain F⇩1⇩3 where ‹❙𝒜[F⇩1⇩3]b & ❙Δ[F⇩1⇩3]b & ¬❙𝒜[F⇩1⇩3]a & ❙Δ[F⇩1⇩3]a›
6180 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6181 AOT_hence ‹❙𝒜[F⇩1⇩3]b› and ‹❙Δ[F⇩1⇩3]b› and ‹¬❙𝒜[F⇩1⇩3]a› and ‹❙Δ[F⇩1⇩3]a›
6182 using "&E" by blast+
6183 note props = props this
6184
6185 let ?Π = "«[λy [O!]y ∨ ¬q⇩0]»"
6186 AOT_modally_strict {
6187 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6188 } note 1 = this
6189 moreover AOT_have ‹❙𝒜[«?Π»]b & ❙Δ[«?Π»]b & ❙𝒜[«?Π»]a & ¬❙Δ[«?Π»]a›
6190 proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6191 AOT_show ‹❙𝒜([O!]b ∨ ¬q⇩0)›
6192 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6193 next AOT_show ‹❙Δ([O!]b ∨ ¬q⇩0)›
6194 by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6195 next AOT_show ‹❙𝒜([O!]a ∨ ¬q⇩0)›
6196 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6197 next AOT_show ‹¬❙Δ([O!]a ∨ ¬q⇩0)›
6198 proof(rule act_and_pos_not_not_delta)
6199 AOT_show ‹❙𝒜([O!]a ∨ ¬q⇩0)›
6200 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6201 next
6202 AOT_have ‹□¬[O!]a›
6203 using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_ord_a not_delta_ord_a "raa-cor:6" by blast
6204 moreover AOT_have ‹◇q⇩0›
6205 by (meson "&E"(1) q⇩0_prop)
6206 ultimately AOT_have 2: ‹◇(¬[O!]a & q⇩0)›
6207 by (metis "KBasic:16" "&I" "vdash-properties:10")
6208 AOT_show ‹◇¬([O!]a ∨ ¬q⇩0)›
6209 proof (AOT_subst_rev ‹«¬[O!]a & q⇩0»› ‹«¬([O!]a ∨ ¬q⇩0)»›)
6210 AOT_modally_strict {
6211 AOT_show ‹¬[O!]a & q⇩0 ≡ ¬([O!]a ∨ ¬q⇩0)›
6212 by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2)
6213 "∨E"(3) "deduction-theorem" "≡I" "raa-cor:3")
6214 }
6215 next
6216 AOT_show ‹◇(¬[O!]a & q⇩0)›
6217 using "2" by blast
6218 qed
6219 qed
6220 qed
6221 ultimately AOT_obtain F⇩1⇩4 where ‹❙𝒜[F⇩1⇩4]b & ❙Δ[F⇩1⇩4]b & ❙𝒜[F⇩1⇩4]a & ¬❙Δ[F⇩1⇩4]a›
6222 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6223 AOT_hence ‹❙𝒜[F⇩1⇩4]b› and ‹❙Δ[F⇩1⇩4]b› and ‹❙𝒜[F⇩1⇩4]a› and ‹¬❙Δ[F⇩1⇩4]a›
6224 using "&E" by blast+
6225 note props = props this
6226
6227 AOT_have ‹[L]↓›
6228 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
6229 moreover AOT_have ‹❙𝒜[L]b & ❙Δ[L]b & ❙𝒜[L]a & ❙Δ[L]a›
6230 proof (safe intro!: "&I")
6231 AOT_show ‹❙𝒜[L]b›
6232 by (meson nec_L "nec-imp-act" "vdash-properties:10")
6233 next AOT_show ‹❙Δ[L]b› using nec_L nec_delta by blast
6234 next AOT_show ‹❙𝒜[L]a› by (meson nec_L "nec-imp-act" "vdash-properties:10")
6235 next AOT_show ‹❙Δ[L]a› using nec_L nec_delta by blast
6236 qed
6237 ultimately AOT_obtain F⇩1⇩5 where ‹❙𝒜[F⇩1⇩5]b & ❙Δ[F⇩1⇩5]b & ❙𝒜[F⇩1⇩5]a & ❙Δ[F⇩1⇩5]a›
6238 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6239 AOT_hence ‹❙𝒜[F⇩1⇩5]b› and ‹❙Δ[F⇩1⇩5]b› and ‹❙𝒜[F⇩1⇩5]a› and ‹❙Δ[F⇩1⇩5]a›
6240 using "&E" by blast+
6241 note props = props this
6242
6243 show ?thesis
6244 by (rule "∃I"(2)[where β=F⇩0]; rule "∃I"(2)[where β=F⇩1]; rule "∃I"(2)[where β=F⇩2];
6245 rule "∃I"(2)[where β=F⇩3]; rule "∃I"(2)[where β=F⇩4]; rule "∃I"(2)[where β=F⇩5];
6246 rule "∃I"(2)[where β=F⇩6]; rule "∃I"(2)[where β=F⇩7]; rule "∃I"(2)[where β=F⇩8];
6247 rule "∃I"(2)[where β=F⇩9]; rule "∃I"(2)[where β=F⇩1⇩0]; rule "∃I"(2)[where β=F⇩1⇩1];
6248 rule "∃I"(2)[where β=F⇩1⇩2]; rule "∃I"(2)[where β=F⇩1⇩3]; rule "∃I"(2)[where β=F⇩1⇩4];
6249 rule "∃I"(2)[where β=F⇩1⇩5]; safe intro!: "&I")
6250 (match conclusion in "[?v ⊨ [F] ≠ [G]]" for F G ⇒ ‹
6251 match props in A: "[?v ⊨ ¬φ{F}]" for φ ⇒ ‹
6252 match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail› ¦ _ ⇒ ‹
6253 match props in B: "[?v ⊨ φ{G}]" ⇒ ‹
6254 fact "pos-not-equiv-ne:4"[where F=F and G=G and φ=φ, THEN "→E",
6255 OF "oth-class-taut:4:h"[THEN "≡E"(2)],
6256 OF "Disjunction Addition"(2)[THEN "→E"],
6257 OF "&I", OF A, OF B]››››)+
6258qed
6259
6260AOT_theorem "o-objects-exist:1": ‹□∃x O!x›
6261proof(rule RN)
6262 AOT_modally_strict {
6263 AOT_obtain a where ‹◇(E!a & ¬❙𝒜[E!]a)›
6264 using "∃E"[rotated, OF "qml:4"[axiom_inst, THEN "BF◇"[THEN "→E"]]] by blast
6265 AOT_hence 1: ‹◇E!a› by (metis "KBasic2:3" "&E"(1) "→E")
6266 AOT_have ‹[λx ◇[E!]x]a›
6267 proof (rule "β←C"(1); "cqt:2[lambda]"?)
6268 AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst] by blast
6269 next
6270 AOT_show ‹◇E!a› by (fact 1)
6271 qed
6272 AOT_hence ‹O!a› by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6273 AOT_thus ‹∃x [O!]x› by (rule "∃I")
6274 }
6275qed
6276
6277AOT_theorem "o-objects-exist:2": ‹□∃x A!x›
6278proof (rule RN)
6279 AOT_modally_strict {
6280 AOT_obtain a where ‹[A!]a›
6281 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
6282 AOT_thus ‹∃x A!x› using "∃I" by blast
6283 }
6284qed
6285
6286AOT_theorem "o-objects-exist:3": ‹□¬∀x O!x›
6287 by (rule RN) (metis (no_types, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(4) "modus-tollens:1" "o-objects-exist:2" "oa-contingent:2" "qml:2"[axiom_inst] "reductio-aa:2")
6288
6289AOT_theorem "o-objects-exist:4": ‹□¬∀x A!x›
6290 by (rule RN) (metis (mono_tags, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(1) "modus-tollens:1" "o-objects-exist:1" "oa-contingent:2" "qml:2"[axiom_inst] "→E")
6291
6292AOT_theorem "o-objects-exist:5": ‹□¬∀x E!x›
6293proof (rule RN; rule "raa-cor:2")
6294 AOT_modally_strict {
6295 AOT_assume ‹∀x E!x›
6296 moreover AOT_obtain a where abs: ‹A!a›
6297 using "o-objects-exist:2"[THEN "qml:2"[axiom_inst, THEN "→E"]] "∃E"[rotated] by blast
6298 ultimately AOT_have ‹E!a› using "∀E" by blast
6299 AOT_hence 1: ‹◇E!a› by (metis "T◇" "→E")
6300 AOT_have ‹[λy ◇E!y]a›
6301 proof (rule "β←C"(1); "cqt:2[lambda]"?)
6302 AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst].
6303 next
6304 AOT_show ‹◇E!a› by (fact 1)
6305 qed
6306 AOT_hence ‹O!a›
6307 by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6308 AOT_hence ‹¬A!a› by (metis "≡E"(1) "oa-contingent:2")
6309 AOT_thus ‹p & ¬p› for p using abs by (metis "raa-cor:3")
6310 }
6311qed
6312
6313AOT_theorem partition: ‹¬∃x (O!x & A!x)›
6314proof(rule "raa-cor:2")
6315 AOT_assume ‹∃x (O!x & A!x)›
6316 then AOT_obtain a where ‹O!a & A!a› using "∃E"[rotated] by blast
6317 AOT_thus ‹p & ¬p› for p by (metis "&E"(1) "Conjunction Simplification"(2) "≡E"(1) "modus-tollens:1" "oa-contingent:2" "raa-cor:3")
6318qed
6319
6320AOT_define eq_E :: ‹Π› ("'(=⇩E')") "=E": ‹(=⇩E) =⇩d⇩f [λxy O!x & O!y & □∀F ([F]x ≡ [F]y)]›
6321
6322syntax "_AOT_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "=⇩E" 50)
6323translations
6324 "_AOT_eq_E_infix κ κ'" == "CONST AOT_exe (CONST eq_E) (CONST Pair κ κ')"
6325
6326print_translation‹
6327AOT_syntax_print_translations
6328[(\<^const_syntax>‹AOT_exe›, fn ctxt => fn [
6329 Const ("\<^const>AOT_PLM.eq_E", _),
6330 Const (\<^const_syntax>‹Pair›, _) $ lhs $ rhs
6331] => Const (\<^syntax_const>‹_AOT_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6332
6333text‹Note: Not explicitly mentioned as theorem in PLM.›
6334AOT_theorem "=E[denotes]": ‹[(=⇩E)]↓›
6335 by (rule "=⇩d⇩fI"(2)[OF "=E"]) "cqt:2[lambda]"+
6336
6337AOT_theorem "=E-simple:1": ‹x =⇩E y ≡ (O!x & O!y & □∀F ([F]x ≡ [F]y))›
6338proof -
6339
6340 AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6341 by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6342 AOT_have 1: ‹[λxy [O!]x & [O!]y & □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6343 show ?thesis apply (rule "=⇩d⇩fI"(2)[OF "=E"]; "cqt:2[lambda]"?)
6344 using "beta-C-meta"[THEN "→E", OF 1, unvarify ν⇩1ν⇩n, of "(AOT_term_of_var x,AOT_term_of_var y)", OF 0]
6345 by fast
6346qed
6347
6348AOT_theorem "=E-simple:2": ‹x =⇩E y → x = y›
6349proof (rule "→I")
6350 AOT_assume ‹x =⇩E y›
6351 AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)› using "=E-simple:1"[THEN "≡E"(1)] by blast
6352 AOT_thus ‹x = y›
6353 using "≡⇩d⇩fI"[OF "identity:1"] "∨I" by blast
6354qed
6355
6356AOT_theorem "id-nec3:1": ‹x =⇩E y ≡ □(x =⇩E y)›
6357proof (rule "≡I"; rule "→I")
6358 AOT_assume ‹x =⇩E y›
6359 AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6360 using "=E-simple:1" "≡E" by blast
6361 AOT_hence ‹□O!x & □O!y & □□∀F ([F]x ≡ [F]y)›
6362 by (metis "S5Basic:6" "&I" "&E"(1) "&E"(2) "≡E"(4) "oa-facts:1" "raa-cor:3" "vdash-properties:10")
6363 AOT_hence 1: ‹□(O!x & O!y & □∀F ([F]x ≡ [F]y))›
6364 by (metis "&E"(1) "&E"(2) "≡E"(2) "KBasic:3" "&I")
6365 AOT_show ‹□(x =⇩E y)›
6366 apply (AOT_subst ‹«x =⇩E y»› ‹«O!x & O!y & □∀F ([F]x ≡ [F]y)»›)
6367 using "=E-simple:1" apply presburger
6368 by (simp add: "1")
6369next
6370 AOT_assume ‹□(x =⇩E y)›
6371 AOT_thus ‹x =⇩E y› using "qml:2"[axiom_inst, THEN "→E"] by blast
6372qed
6373
6374AOT_theorem "id-nec3:2": ‹◇(x =⇩E y) ≡ x =⇩E y›
6375 by (meson "RE◇" "S5Basic:2" "id-nec3:1" "≡E"(1) "≡E"(5) "Commutativity of ≡")
6376
6377AOT_theorem "id-nec3:3": ‹◇(x =⇩E y) ≡ □(x =⇩E y)›
6378 by (meson "id-nec3:1" "id-nec3:2" "≡E"(5))
6379
6380syntax "_AOT_non_eq_E" :: ‹Π› ("'(≠⇩E')")
6381translations
6382 (Π) "(≠⇩E)" == (Π) "(=⇩E)⇧-"
6383syntax "_AOT_non_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "≠⇩E" 50)
6384translations
6385 "_AOT_non_eq_E_infix κ κ'" == "CONST AOT_exe (CONST relation_negation (CONST eq_E)) (CONST Pair κ κ')"
6386
6387print_translation‹
6388AOT_syntax_print_translations
6389[(\<^const_syntax>‹AOT_exe›, fn ctxt => fn [
6390 Const (\<^const_syntax>‹relation_negation›, _) $ Const ("\<^const>AOT_PLM.eq_E", _),
6391 Const (\<^const_syntax>‹Pair›, _) $ lhs $ rhs
6392] => Const (\<^syntax_const>‹_AOT_non_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6393AOT_theorem "thm-neg=E": ‹x ≠⇩E y ≡ ¬(x =⇩E y)›
6394proof -
6395
6396 AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6397 by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6398 AOT_have θ: ‹[λx⇩1...x⇩2 ¬(=⇩E)x⇩1...x⇩2]↓› by "cqt:2[lambda]"
6399 AOT_have ‹x ≠⇩E y ≡ [λx⇩1...x⇩2 ¬(=⇩E)x⇩1...x⇩2]xy›
6400 by (rule "=⇩d⇩fI"(1)[OF "df-relation-negation", OF θ])
6401 (meson "oth-class-taut:3:a")
6402 also AOT_have ‹… ≡ ¬(=⇩E)xy›
6403 apply (rule "beta-C-meta"[THEN "→E", unvarify ν⇩1ν⇩n])
6404 apply "cqt:2[lambda]"
6405 by (fact 0)
6406 finally show ?thesis.
6407qed
6408
6409AOT_theorem "id-nec4:1": ‹x ≠⇩E y ≡ □(x ≠⇩E y)›
6410proof -
6411 AOT_have ‹x ≠⇩E y ≡ ¬(x =⇩E y)› using "thm-neg=E".
6412 also AOT_have ‹… ≡ ¬◇(x =⇩E y)›
6413 by (meson "id-nec3:2" "≡E"(1) "Commutativity of ≡" "oth-class-taut:4:b")
6414 also AOT_have ‹… ≡ □¬(x =⇩E y)›
6415 by (meson "KBasic2:1" "≡E"(2) "Commutativity of ≡")
6416 also AOT_have ‹… ≡ □(x ≠⇩E y)›
6417 by (AOT_subst_rev "«x ≠⇩E y»" "«¬(x =⇩E y)»")
6418 (auto simp: "thm-neg=E" "oth-class-taut:3:a")
6419 finally show ?thesis.
6420qed
6421
6422AOT_theorem "id-nec4:2": ‹◇(x ≠⇩E y) ≡ (x ≠⇩E y)›
6423 by (meson "RE◇" "S5Basic:2" "id-nec4:1" "≡E"(2) "≡E"(5) "Commutativity of ≡")
6424
6425AOT_theorem "id-nec4:3": ‹◇(x ≠⇩E y) ≡ □(x ≠⇩E y)›
6426 by (meson "id-nec4:1" "id-nec4:2" "≡E"(5))
6427
6428AOT_theorem "id-act2:1": ‹x =⇩E y ≡ ❙𝒜x =⇩E y›
6429 by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec3:2" "≡E"(1) "≡E"(6))
6430AOT_theorem "id-act2:2": ‹x ≠⇩E y ≡ ❙𝒜x ≠⇩E y›
6431 by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec4:2" "≡E"(1) "≡E"(6))
6432
6433AOT_theorem "ord=Eequiv:1": ‹O!x → x =⇩E x›
6434proof (rule "→I")
6435 AOT_assume 1: ‹O!x›
6436 AOT_show ‹x =⇩E x›
6437 apply (rule "=⇩d⇩fI"(2)[OF "=E"]) apply "cqt:2[lambda]"
6438 apply (rule "β←C"(1))
6439 apply "cqt:2[lambda]"
6440 apply (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6441 by (simp add: "1" RN "&I" "oth-class-taut:3:a" "universal-cor")
6442qed
6443
6444AOT_theorem "ord=Eequiv:2": ‹x =⇩E y → y =⇩E x›
6445proof(rule CP)
6446 AOT_assume 1: ‹x =⇩E y›
6447 AOT_hence 2: ‹x = y› by (metis "=E-simple:2" "vdash-properties:10")
6448 AOT_have ‹O!x› using 1 by (meson "&E"(1) "=E-simple:1" "≡E"(1))
6449 AOT_hence ‹x =⇩E x› using "ord=Eequiv:1" "→E" by blast
6450 AOT_thus ‹y =⇩E x› using "rule=E"[rotated, OF 2] by fast
6451qed
6452
6453AOT_theorem "ord=Eequiv:3": ‹(x =⇩E y & y =⇩E z) → x =⇩E z›
6454proof (rule CP)
6455 AOT_assume 1: ‹x =⇩E y & y =⇩E z›
6456 AOT_hence ‹x = y & y = z›
6457 by (metis "&I" "&E"(1) "&E"(2) "=E-simple:2" "vdash-properties:6")
6458 AOT_hence ‹x = z› by (metis "id-eq:3" "vdash-properties:6")
6459 moreover AOT_have ‹x =⇩E x›
6460 using 1[THEN "&E"(1)] "&E"(1) "=E-simple:1" "≡E"(1) "ord=Eequiv:1" "→E" by blast
6461 ultimately AOT_show ‹x =⇩E z›
6462 using "rule=E" by fast
6463qed
6464
6465AOT_theorem "ord-=E=:1": ‹(O!x ∨ O!y) → □(x = y ≡ x =⇩E y)›
6466proof(rule CP)
6467 AOT_assume ‹O!x ∨ O!y›
6468 moreover {
6469 AOT_assume ‹O!x›
6470 AOT_hence ‹□O!x› by (metis "oa-facts:1" "vdash-properties:10")
6471 moreover {
6472 AOT_modally_strict {
6473 AOT_have ‹O!x → (x = y ≡ x =⇩E y)›
6474 proof (rule "→I"; rule "≡I"; rule "→I")
6475 AOT_assume ‹O!x›
6476 AOT_hence ‹x =⇩E x› by (metis "ord=Eequiv:1" "→E")
6477 moreover AOT_assume ‹x = y›
6478 ultimately AOT_show ‹x =⇩E y› using "rule=E" by fast
6479 next
6480 AOT_assume ‹x =⇩E y›
6481 AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6482 qed
6483 }
6484 AOT_hence ‹□O!x → □(x = y ≡ x =⇩E y)› by (metis "RM:1")
6485 }
6486 ultimately AOT_have ‹□(x = y ≡ x =⇩E y)› using "→E" by blast
6487 }
6488 moreover {
6489 AOT_assume ‹O!y›
6490 AOT_hence ‹□O!y› by (metis "oa-facts:1" "vdash-properties:10")
6491 moreover {
6492 AOT_modally_strict {
6493 AOT_have ‹O!y → (x = y ≡ x =⇩E y)›
6494 proof (rule "→I"; rule "≡I"; rule "→I")
6495 AOT_assume ‹O!y›
6496 AOT_hence ‹y =⇩E y› by (metis "ord=Eequiv:1" "→E")
6497 moreover AOT_assume ‹x = y›
6498 ultimately AOT_show ‹x =⇩E y› using "rule=E" id_sym by fast
6499 next
6500 AOT_assume ‹x =⇩E y›
6501 AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6502 qed
6503 }
6504 AOT_hence ‹□O!y → □(x = y ≡ x =⇩E y)› by (metis "RM:1")
6505 }
6506 ultimately AOT_have ‹□(x = y ≡ x =⇩E y)› using "→E" by blast
6507 }
6508 ultimately AOT_show ‹□(x = y ≡ x =⇩E y)› by (metis "∨E"(3) "raa-cor:1")
6509qed
6510
6511AOT_theorem "ord-=E=:2": ‹O!y → [λx x = y]↓›
6512proof (rule "→I"; rule "safe-ext"[axiom_inst, THEN "→E"]; rule "&I")
6513 AOT_show ‹[λx x =⇩E y]↓› by "cqt:2[lambda]"
6514next
6515 AOT_assume ‹O!y›
6516 AOT_hence 1: ‹□(x = y ≡ x =⇩E y)› for x using "ord-=E=:1" "→E" "∨I" by blast
6517 AOT_have ‹□(x =⇩E y ≡ x = y)› for x
6518 by (AOT_subst ‹«x =⇩E y ≡ x = y»› ‹«x = y ≡ x =⇩E y»›)
6519 (auto simp add: "Commutativity of ≡" 1)
6520 AOT_hence ‹∀x □(x =⇩E y ≡ x = y)› by (rule GEN)
6521 AOT_thus ‹□∀x (x =⇩E y ≡ x = y)› by (rule BF[THEN "→E"])
6522qed
6523
6524
6525AOT_theorem "ord-=E=:3": ‹[λxy O!x & O!y & x = y]↓›
6526proof (rule "safe-ext[2]"[axiom_inst, THEN "→E"]; rule "&I")
6527 AOT_show ‹[λxy O!x & O!y & x =⇩E y]↓› by "cqt:2[lambda]"
6528next
6529 AOT_show ‹□∀x∀y ([O!]x & [O!]y & x =⇩E y ≡ [O!]x & [O!]y & x = y)›
6530 proof (rule RN; rule GEN; rule GEN; rule "≡I"; rule "→I")
6531 AOT_modally_strict {
6532 AOT_show ‹[O!]x & [O!]y & x = y› if ‹[O!]x & [O!]y & x =⇩E y› for x y
6533 by (metis "&I" "&E"(1) "Conjunction Simplification"(2) "=E-simple:2"
6534 "modus-tollens:1" "raa-cor:1" that)
6535 }
6536 next
6537 AOT_modally_strict {
6538 AOT_show ‹[O!]x & [O!]y & x =⇩E y› if ‹[O!]x & [O!]y & x = y› for x y
6539 apply(safe intro!: "&I")
6540 apply (metis that[THEN "&E"(1), THEN "&E"(1)])
6541 apply (metis that[THEN "&E"(1), THEN "&E"(2)])
6542 using "rule=E"[rotated, OF that[THEN "&E"(2)]]
6543 "ord=Eequiv:1"[THEN "→E", OF that[THEN "&E"(1), THEN "&E"(1)]] by fast
6544 }
6545 qed
6546qed
6547
6548AOT_theorem "ind-nec": ‹∀F ([F]x ≡ [F]y) → □∀F ([F]x ≡ [F]y)›
6549proof(rule "→I")
6550 AOT_assume ‹∀F ([F]x ≡ [F]y)›
6551 moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6552 ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x ≡ [λx □∀F ([F]x ≡ [F]y)]y›
6553 using "∀E" by blast
6554 moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]y›
6555 apply (rule "β←C"(1))
6556 apply "cqt:2[lambda]"
6557 apply (fact "cqt:2[const_var]"[axiom_inst])
6558 by (simp add: RN GEN "oth-class-taut:3:a")
6559 ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x› using "≡E" by blast
6560 AOT_thus ‹□∀F ([F]x ≡ [F]y)›
6561 using "β→C"(1) by blast
6562qed
6563
6564AOT_theorem "ord=E:1": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x =⇩E y)›
6565proof (rule "→I"; rule "→I")
6566 AOT_assume ‹∀F ([F]x ≡ [F]y)›
6567 AOT_hence ‹□∀F ([F]x ≡ [F]y)›
6568 using "ind-nec"[THEN "→E"] by blast
6569 moreover AOT_assume ‹O!x & O!y›
6570 ultimately AOT_have ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6571 using "&I" by blast
6572 AOT_thus ‹x =⇩E y› using "=E-simple:1"[THEN "≡E"(2)] by blast
6573qed
6574
6575AOT_theorem "ord=E:2": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x = y)›
6576proof (rule "→I"; rule "→I")
6577 AOT_assume ‹O!x & O!y›
6578 moreover AOT_assume ‹∀F ([F]x ≡ [F]y)›
6579 ultimately AOT_have ‹x =⇩E y›
6580 using "ord=E:1" "→E" by blast
6581 AOT_thus ‹x = y› using "=E-simple:2"[THEN "→E"] by blast
6582qed
6583
6584AOT_theorem "ord=E2:1": ‹(O!x & O!y) → (x ≠ y ≡ [λz z =⇩E x] ≠ [λz z =⇩E y])›
6585proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡⇩d⇩fI"[OF "=-infix"]; rule "raa-cor:2")
6586 AOT_assume 0: ‹O!x & O!y›
6587 AOT_assume ‹x ≠ y›
6588 AOT_hence 1: ‹¬(x = y)› using "≡⇩d⇩fE"[OF "=-infix"] by blast
6589 AOT_assume ‹[λz z =⇩E x] = [λz z =⇩E y]›
6590 moreover AOT_have ‹[λz z =⇩E x]x›
6591 apply (rule "β←C"(1))
6592 apply "cqt:2[lambda]"
6593 apply (fact "cqt:2[const_var]"[axiom_inst])
6594 using "ord=Eequiv:1"[THEN "→E", OF 0[THEN "&E"(1)]].
6595 ultimately AOT_have ‹[λz z =⇩E y]x› using "rule=E" by fast
6596 AOT_hence ‹x =⇩E y› using "β→C"(1) by blast
6597 AOT_hence ‹x = y› by (metis "=E-simple:2" "vdash-properties:6")
6598 AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6599next
6600 AOT_assume ‹[λz z =⇩E x] ≠ [λz z =⇩E y]›
6601 AOT_hence 0: ‹¬([λz z =⇩E x] = [λz z =⇩E y])› using "≡⇩d⇩fE"[OF "=-infix"] by blast
6602 AOT_have ‹[λz z =⇩E x]↓› by "cqt:2[lambda]"
6603 AOT_hence ‹[λz z =⇩E x] = [λz z =⇩E x]›
6604 by (metis "rule=I:1")
6605 moreover AOT_assume ‹x = y›
6606 ultimately AOT_have ‹[λz z =⇩E x] = [λz z =⇩E y]›
6607 using "rule=E" by fast
6608 AOT_thus ‹[λz z =⇩E x] = [λz z =⇩E y] & ¬([λz z =⇩E x] = [λz z =⇩E y])›
6609 using 0 "&I" by blast
6610qed
6611
6612AOT_theorem "ord=E2:2": ‹(O!x & O!y) → (x ≠ y ≡ [λz z = x] ≠ [λz z = y])›
6613proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡⇩d⇩fI"[OF "=-infix"]; rule "raa-cor:2")
6614 AOT_assume 0: ‹O!x & O!y›
6615 AOT_assume ‹x ≠ y›
6616 AOT_hence 1: ‹¬(x = y)› using "≡⇩d⇩fE"[OF "=-infix"] by blast
6617 AOT_assume ‹[λz z = x] = [λz z = y]›
6618 moreover AOT_have ‹[λz z = x]x›
6619 apply (rule "β←C"(1))
6620 apply (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6621 apply (fact "cqt:2[const_var]"[axiom_inst])
6622 by (simp add: "id-eq:1")
6623 ultimately AOT_have ‹[λz z = y]x› using "rule=E" by fast
6624 AOT_hence ‹x = y› using "β→C"(1) by blast
6625 AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6626next
6627 AOT_assume 0: ‹O!x & O!y›
6628 AOT_assume ‹[λz z = x] ≠ [λz z = y]›
6629 AOT_hence 1: ‹¬([λz z = x] = [λz z = y])› using "≡⇩d⇩fE"[OF "=-infix"] by blast
6630 AOT_have ‹[λz z = x]↓› by (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6631 AOT_hence ‹[λz z = x] = [λz z = x]›
6632 by (metis "rule=I:1")
6633 moreover AOT_assume ‹x = y›
6634 ultimately AOT_have ‹[λz z = x] = [λz z = y]›
6635 using "rule=E" by fast
6636 AOT_thus ‹[λz z = x] = [λz z = y] & ¬([λz z = x] = [λz z = y])›
6637 using 1 "&I" by blast
6638qed
6639
6640AOT_theorem ordnecfail: ‹O!x → □¬∃F x[F]›
6641 by (meson "RM:1" "deduction-theorem" nocoder "oa-facts:1" "vdash-properties:10" "vdash-properties:1[2]")
6642
6643AOT_theorem "ab-obey:1": ‹(A!x & A!y) → (∀F (x[F] ≡ y[F]) → x = y)›
6644proof (rule "→I"; rule "→I")
6645 AOT_assume 1: ‹A!x & A!y›
6646 AOT_assume ‹∀F (x[F] ≡ y[F])›
6647 AOT_hence ‹x[F] ≡ y[F]› for F using "∀E" by blast
6648 AOT_hence ‹□(x[F] ≡ y[F])› for F by (metis "en-eq:6[1]" "≡E"(1))
6649 AOT_hence ‹∀F □(x[F] ≡ y[F])› by (rule GEN)
6650 AOT_hence ‹□∀F (x[F] ≡ y[F])› by (rule BF[THEN "→E"])
6651 AOT_thus ‹x = y›
6652 using "≡⇩d⇩fI"[OF "identity:1", OF "∨I"(2)] 1 "&I" by blast
6653qed
6654
6655AOT_theorem "ab-obey:2": ‹(∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])) → x ≠ y›
6656proof (rule "→I"; rule "≡⇩d⇩fI"[OF "=-infix"]; rule "raa-cor:2")
6657 AOT_assume 1: ‹x = y›
6658 AOT_assume ‹∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])›
6659 moreover {
6660 AOT_assume ‹∃F (x[F] & ¬y[F])›
6661 then AOT_obtain F where ‹x[F] & ¬y[F]› using "∃E"[rotated] by blast
6662 moreover AOT_have ‹y[F]› using calculation[THEN "&E"(1)] 1 "rule=E" by fast
6663 ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(2) "modus-tollens:2" "raa-cor:3")
6664 }
6665 moreover {
6666 AOT_assume ‹∃F (y[F] & ¬x[F])›
6667 then AOT_obtain F where ‹y[F] & ¬x[F]› using "∃E"[rotated] by blast
6668 moreover AOT_have ‹¬y[F]› using calculation[THEN "&E"(2)] 1 "rule=E" by fast
6669 ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(1) "modus-tollens:1" "raa-cor:3")
6670 }
6671 ultimately AOT_show ‹p & ¬p› for p by (metis "∨E"(3) "raa-cor:1")
6672qed
6673
6674AOT_theorem "encoders-are-abstract": ‹∃F x[F] → A!x›
6675 by (meson "deduction-theorem" "≡E"(2) "modus-tollens:2" nocoder
6676 "oa-contingent:3" "vdash-properties:1[2]")
6677
6678AOT_theorem "denote=:1": ‹∀H∃x x[H]›
6679 by (rule GEN; rule "existence:2[1]"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6680
6681AOT_theorem "denote=:2": ‹∀G∃x⇩1...∃x⇩n x⇩1...x⇩n[H]›
6682 by (rule GEN; rule "existence:2"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6683
6684AOT_theorem "denote=:2[2]": ‹∀G∃x⇩1∃x⇩2 x⇩1x⇩2[H]›
6685 by (rule GEN; rule "existence:2[2]"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6686
6687AOT_theorem "denote=:2[3]": ‹∀G∃x⇩1∃x⇩2∃x⇩3 x⇩1x⇩2x⇩3[H]›
6688 by (rule GEN; rule "existence:2[3]"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6689
6690AOT_theorem "denote=:2[4]": ‹∀G∃x⇩1∃x⇩2∃x⇩3∃x⇩4 x⇩1x⇩2x⇩3x⇩4[H]›
6691 by (rule GEN; rule "existence:2[4]"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6692
6693AOT_theorem "denote=:3": ‹∃x x[Π] ≡ ∃H (H = Π)›
6694 using "existence:2[1]" "free-thms:1" "≡E"(2) "≡E"(5) "Commutativity of ≡" "≡Df" by blast
6695
6696AOT_theorem "denote=:4": ‹(∃x⇩1...∃x⇩n x⇩1...x⇩n[Π]) ≡ ∃H (H = Π)›
6697 using "existence:2" "free-thms:1" "≡E"(6) "≡Df" by blast
6698
6699AOT_theorem "denote=:4[2]": ‹(∃x⇩1∃x⇩2 x⇩1x⇩2[Π]) ≡ ∃H (H = Π)›
6700 using "existence:2[2]" "free-thms:1" "≡E"(6) "≡Df" by blast
6701
6702AOT_theorem "denote=:4[3]": ‹(∃x⇩1∃x⇩2∃x⇩3 x⇩1x⇩2x⇩3[Π]) ≡ ∃H (H = Π)›
6703 using "existence:2[3]" "free-thms:1" "≡E"(6) "≡Df" by blast
6704
6705AOT_theorem "denote=:4[4]": ‹(∃x⇩1∃x⇩2∃x⇩3∃x⇩4 x⇩1x⇩2x⇩3x⇩4[Π]) ≡ ∃H (H = Π)›
6706 using "existence:2[4]" "free-thms:1" "≡E"(6) "≡Df" by blast
6707
6708AOT_theorem "A-objects!": ‹∃!x (A!x & ∀F (x[F] ≡ φ{F}))›
6709proof (rule "uniqueness:1"[THEN "≡⇩d⇩fI"])
6710 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ φ{F})›
6711 using "A-objects"[axiom_inst] "∃E"[rotated] by blast
6712 AOT_have ‹(A!β & ∀F (β[F] ≡ φ{F})) → β = a› for β
6713 proof (rule "→I")
6714 AOT_assume β_prop: ‹[A!]β & ∀F (β[F] ≡ φ{F})›
6715 AOT_hence ‹β[F] ≡ φ{F}› for F using "∀E" "&E" by blast
6716 AOT_hence ‹β[F] ≡ a[F]› for F
6717 using a_prop[THEN "&E"(2)] "∀E" "≡E"(2) "≡E"(5) "Commutativity of ≡" by fast
6718 AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6719 AOT_thus ‹β = a›
6720 using "ab-obey:1"[THEN "→E", OF "&I"[OF β_prop[THEN "&E"(1)], OF a_prop[THEN "&E"(1)]], THEN "→E"] by blast
6721 qed
6722 AOT_hence ‹∀β ((A!β & ∀F (β[F] ≡ φ{F})) → β = a)› by (rule GEN)
6723 AOT_thus ‹∃α ([A!]α & ∀F (α[F] ≡ φ{F}) & ∀β ([A!]β & ∀F (β[F] ≡ φ{F}) → β = α)) ›
6724 using "∃I" using a_prop "&I" by fast
6725qed
6726
6727AOT_theorem "obj-oth:1": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y))›
6728 using "A-objects!" by fast
6729
6730AOT_theorem "obj-oth:2": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y & [F]z))›
6731 using "A-objects!" by fast
6732
6733AOT_theorem "obj-oth:3": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y ∨ [F]z))›
6734 using "A-objects!" by fast
6735
6736AOT_theorem "obj-oth:4": ‹∃!x (A!x & ∀F (x[F] ≡ □[F]y))›
6737 using "A-objects!" by fast
6738
6739AOT_theorem "obj-oth:5": ‹∃!x (A!x & ∀F (x[F] ≡ F = G))›
6740 using "A-objects!" by fast
6741
6742AOT_theorem "obj-oth:6": ‹∃!x (A!x & ∀F (x[F] ≡ □∀y([G]y → [F]y)))›
6743 using "A-objects!" by fast
6744
6745AOT_theorem "A-descriptions": ‹❙ιx (A!x & ∀F (x[F] ≡ φ{F}))↓›
6746 by (rule "A-Exists:2"[THEN "≡E"(2)]; rule "RA[2]"; rule "A-objects!")
6747
6748AOT_act_theorem "thm-can-terms2": ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6749 using "y-in:2" by blast
6750
6751AOT_theorem "can-ab2": ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F})) → A!y›
6752proof(rule "→I")
6753 AOT_assume ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6754 AOT_hence ‹❙𝒜(A!y & ∀F (y[F] ≡ φ{F}))›
6755 using "actual-desc:2"[THEN "→E"] by blast
6756 AOT_hence ‹❙𝒜A!y› by (metis "Act-Basic:2" "&E"(1) "≡E"(1))
6757 AOT_thus ‹A!y› by (metis "≡E"(2) "oa-facts:8")
6758qed
6759
6760AOT_act_theorem "desc-encode": ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6761proof -
6762 AOT_have ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6763 by (simp add: "A-descriptions")
6764 AOT_hence ‹A!❙ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6765 using "y-in:3"[THEN "→E"] by blast
6766 AOT_thus ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6767 using "&E" "∀E" by blast
6768qed
6769
6770AOT_theorem "desc-nec-encode": ‹❙ιx (A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ ❙𝒜φ{G}›
6771proof -
6772 AOT_have 0: ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6773 by (simp add: "A-descriptions")
6774 AOT_hence ‹❙𝒜(A!❙ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}))›
6775 using "actual-desc:4"[THEN "→E"] by blast
6776 AOT_hence ‹❙𝒜∀F (❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6777 using "Act-Basic:2" "&E"(2) "≡E"(1) by blast
6778 AOT_hence ‹∀F ❙𝒜(❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6779 using "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]" by blast
6780 AOT_hence ‹❙𝒜(❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G})›
6781 using "∀E" by blast
6782 AOT_hence ‹❙𝒜❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ ❙𝒜φ{G}›
6783 using "Act-Basic:5" "≡E"(1) by blast
6784 AOT_thus ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ ❙𝒜φ{G}›
6785 using "en-eq:10[1]"[unvarify x⇩1, OF 0] "≡E"(6) by blast
6786qed
6787
6788AOT_theorem "Box-desc-encode:1": ‹□φ{G} → ❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]›
6789 by (rule "→I"; rule "desc-nec-encode"[THEN "≡E"(2)])
6790 (meson "nec-imp-act" "vdash-properties:10")
6791
6792AOT_theorem "Box-desc-encode:2": ‹□φ{G} → □(❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6793proof(rule CP)
6794 AOT_assume ‹□φ{G}›
6795 AOT_hence ‹□□φ{G}› by (metis "S5Basic:6" "≡E"(1))
6796 moreover AOT_have ‹□□φ{G} → □(❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6797 proof (rule RM; rule "→I")
6798 AOT_modally_strict {
6799 AOT_assume 1: ‹□φ{G}›
6800 AOT_hence ‹❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]› using "Box-desc-encode:1" "→E" by blast
6801 moreover AOT_have ‹φ{G}› using 1 by (meson "qml:2" "vdash-properties:10" "vdash-properties:1[2]")
6802 ultimately AOT_show ‹❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G}›
6803 using "deduction-theorem" "≡I" by simp
6804 }
6805 qed
6806 ultimately AOT_show ‹□(❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})› using "→E" by blast
6807qed
6808
6809definition rigid_condition where ‹rigid_condition φ ≡ ∀v . [v ⊨ ∀α (φ{α} → □φ{α})]›
6810syntax rigid_condition :: ‹id_position ⇒ AOT_prop› ("RIGID'_CONDITION'(_')")
6811
6812AOT_theorem "strict-can:1[E]": assumes ‹RIGID_CONDITION(φ)›
6813 shows ‹∀α (φ{α} → □φ{α})›
6814 using assms[unfolded rigid_condition_def] by auto
6815
6816AOT_theorem "strict-can:1[I]":
6817 assumes ‹❙⊢⇩□ ∀α (φ{α} → □φ{α})›
6818 shows ‹RIGID_CONDITION(φ)›
6819 using assms rigid_condition_def by auto
6820
6821AOT_theorem "box-phi-a:1": assumes ‹RIGID_CONDITION(φ)›
6822 shows ‹(A!x & ∀F (x[F] ≡ φ{F})) → □(A!x & ∀F (x[F] ≡ φ{F}))›
6823proof (rule "→I")
6824 AOT_assume a: ‹A!x & ∀F (x[F] ≡ φ{F})›
6825 AOT_hence b: ‹□A!x› by (metis "Conjunction Simplification"(1) "oa-facts:2" "vdash-properties:10")
6826 AOT_have ‹x[F] ≡ φ{F}› for F using a[THEN "&E"(2)] "∀E" by blast
6827 moreover AOT_have ‹□(x[F] → □x[F])› for F by (meson "pre-en-eq:1[1]" RN)
6828 moreover AOT_have ‹□(φ{F} → □φ{F})› for F using RN "strict-can:1[E]"[OF assms] "∀E" by blast
6829 ultimately AOT_have ‹□(x[F] ≡ φ{F})› for F
6830 by (metis "&I" "sc-eq-box-box:5" "vdash-properties:6")
6831 AOT_hence ‹∀F □(x[F] ≡ φ{F})› by (rule GEN)
6832 AOT_hence ‹□∀F (x[F] ≡ φ{F})› by (rule BF[THEN "→E"])
6833 AOT_thus ‹□([A!]x & ∀F (x[F] ≡ φ{F}))›
6834 using b "KBasic:3" "≡S"(1) "≡E"(2) by blast
6835qed
6836
6837AOT_theorem "box-phi-a:2": assumes ‹RIGID_CONDITION(φ)›
6838 shows ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6839proof(rule "→I")
6840 AOT_assume ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6841 AOT_hence ‹❙𝒜(A!y & ∀F (y[F] ≡ φ{F}))› using "actual-desc:2"[THEN "→E"] by fast
6842 AOT_hence abs: ‹❙𝒜A!y› and ‹❙𝒜∀F (y[F] ≡ φ{F})›
6843 using "Act-Basic:2" "&E" "≡E"(1) by blast+
6844 AOT_hence ‹∀F ❙𝒜(y[F] ≡ φ{F})› by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
6845 AOT_hence ‹❙𝒜(y[F] ≡ φ{F})› for F using "∀E" by blast
6846 AOT_hence ‹❙𝒜y[F] ≡ ❙𝒜φ{F}› for F by (metis "Act-Basic:5" "≡E"(1))
6847 AOT_hence ‹y[F] ≡ φ{F}› for F
6848 using "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6849 by (metis "en-eq:10[1]" "≡E"(6))
6850 AOT_hence ‹∀F (y[F] ≡ φ{F})› by (rule GEN)
6851 AOT_thus ‹[A!]y & ∀F (y[F] ≡ φ{F})› using abs "&I" "≡E"(2) "oa-facts:8" by blast
6852qed
6853
6854AOT_theorem "box-phi-a:3": assumes ‹RIGID_CONDITION(φ)›
6855 shows ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6856 using "desc-nec-encode"
6857 "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=G], THEN RN]]
6858 "≡E"(5) by blast
6859
6860AOT_define Null :: ‹τ ⇒ φ› ("Null'(_')")
6861 "df-null-uni:1": ‹Null(x) ≡⇩d⇩f A!x & ¬∃F x[F]›
6862
6863AOT_define Universal :: ‹τ ⇒ φ› ("Universal'(_')")
6864 "df-null-uni:2": ‹Universal(x) ≡⇩d⇩f A!x & ∀F x[F]›
6865
6866AOT_theorem "null-uni-uniq:1": ‹∃!x Null(x)›
6867proof (rule "uniqueness:1"[THEN "≡⇩d⇩fI"])
6868 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ¬(F = F))›
6869 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6870 AOT_have a_null: ‹¬a[F]› for F
6871 proof (rule "raa-cor:2")
6872 AOT_assume ‹a[F]›
6873 AOT_hence ‹¬(F = F)› using a_prop[THEN "&E"(2)] "∀E" "≡E" by blast
6874 AOT_hence ‹F = F & ¬(F = F)› by (metis "id-eq:1" "raa-cor:3")
6875 AOT_thus ‹p & ¬p› for p by (metis "raa-cor:1")
6876 qed
6877 AOT_have ‹Null(a) & ∀β (Null(β) → β = a)›
6878 proof (rule "&I")
6879 AOT_have ‹¬∃F a[F]› using a_null by (metis "instantiation" "reductio-aa:1")
6880 AOT_thus ‹Null(a)›
6881 using "df-null-uni:1"[THEN "≡⇩d⇩fI"] a_prop[THEN "&E"(1)] "&I" by metis
6882 next
6883 AOT_show ‹∀β (Null(β) → β = a)›
6884 proof (rule GEN; rule "→I")
6885 fix β
6886 AOT_assume a: ‹Null(β)›
6887 AOT_hence ‹¬∃F β[F]›
6888 using "df-null-uni:1"[THEN "≡⇩d⇩fE"] "&E" by blast
6889 AOT_hence β_null: ‹¬β[F]› for F by (metis "existential:2[const_var]" "reductio-aa:1")
6890 AOT_have ‹∀F (β[F] ≡ a[F])›
6891 apply (rule GEN; rule "≡I"; rule CP)
6892 using "raa-cor:3" β_null a_null by blast+
6893 moreover AOT_have ‹A!β› using a "df-null-uni:1"[THEN "≡⇩d⇩fE"] "&E" by blast
6894 ultimately AOT_show ‹β = a›
6895 using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" by blast
6896 qed
6897 qed
6898 AOT_thus ‹∃α (Null(α) & ∀β (Null(β) → β = α))› using "∃I"(2) by fast
6899qed
6900
6901AOT_theorem "null-uni-uniq:2": ‹∃!x Universal(x)›
6902proof (rule "uniqueness:1"[THEN "≡⇩d⇩fI"])
6903 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ F = F)›
6904 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6905 AOT_hence aF: ‹a[F]› for F using "&E" "∀E" "≡E" "id-eq:1" by fast
6906 AOT_hence ‹Universal(a)›
6907 using "df-null-uni:2"[THEN "≡⇩d⇩fI"] "&I" a_prop[THEN "&E"(1)] GEN by blast
6908 moreover AOT_have ‹∀β (Universal(β) → β = a)›
6909 proof (rule GEN; rule "→I")
6910 fix β
6911 AOT_assume ‹Universal(β)›
6912 AOT_hence abs_β: ‹A!β› and ‹β[F]› for F using "df-null-uni:2"[THEN "≡⇩d⇩fE"] "&E" "∀E" by blast+
6913 AOT_hence ‹β[F] ≡ a[F]› for F using aF by (metis "deduction-theorem" "≡I")
6914 AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6915 AOT_thus ‹β = a›
6916 using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" abs_β by blast
6917 qed
6918 ultimately AOT_show ‹∃α (Universal(α) & ∀β (Universal(β) → β = α))›
6919 using "&I" "∃I" by fast
6920qed
6921
6922AOT_theorem "null-uni-uniq:3": ‹❙ιx Null(x)↓›
6923 using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:1" by blast
6924
6925AOT_theorem "null-uni-uniq:4": ‹❙ιx Universal(x)↓›
6926 using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:2" by blast
6927
6928AOT_define Null_object :: ‹κ⇩s› (‹a⇩∅›)
6929 "df-null-uni-terms:1": ‹a⇩∅ =⇩d⇩f ❙ιx Null(x)›
6930
6931AOT_define Universal_object :: ‹κ⇩s› (‹a⇩V›)
6932 "df-null-uni-terms:2": ‹a⇩V =⇩d⇩f ❙ιx Universal(x)›
6933
6934AOT_theorem "null-uni-facts:1": ‹Null(x) → □Null(x)›
6935proof (rule "→I")
6936 AOT_assume ‹Null(x)›
6937 AOT_hence x_abs: ‹A!x› and x_null: ‹¬∃F x[F]›
6938 using "df-null-uni:1"[THEN "≡⇩d⇩fE"] "&E" by blast+
6939 AOT_have ‹¬x[F]› for F using x_null
6940 using "existential:2[const_var]" "reductio-aa:1"
6941 by metis
6942 AOT_hence ‹□¬x[F]› for F by (metis "en-eq:7[1]" "≡E"(1))
6943 AOT_hence ‹∀F □¬x[F]› by (rule GEN)
6944 AOT_hence ‹□∀F ¬x[F]› by (rule BF[THEN "→E"])
6945 moreover AOT_have ‹□∀F ¬x[F] → □¬∃F x[F]›
6946 apply (rule RM)
6947 by (metis (full_types) "instantiation" "cqt:2[const_var]" "deduction-theorem"
6948 "reductio-aa:1" "rule-ui:1" "vdash-properties:1[2]")
6949 ultimately AOT_have ‹□¬∃F x[F]›
6950 by (metis "→E")
6951 moreover AOT_have ‹□A!x› using x_abs
6952 using "oa-facts:2" "vdash-properties:10" by blast
6953 ultimately AOT_have r: ‹□(A!x & ¬∃F x[F])›
6954 by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
6955 AOT_show ‹□Null(x)›
6956 by (AOT_subst "«Null(x)»" "«A!x & ¬∃F x[F]»")
6957 (auto simp: "df-null-uni:1" "≡Df" r)
6958qed
6959
6960AOT_theorem "null-uni-facts:2": ‹Universal(x) → □Universal(x)›
6961proof (rule "→I")
6962 AOT_assume ‹Universal(x)›
6963 AOT_hence x_abs: ‹A!x› and x_univ: ‹∀F x[F]›
6964 using "df-null-uni:2"[THEN "≡⇩d⇩fE"] "&E" by blast+
6965 AOT_have ‹x[F]› for F using x_univ "∀E" by blast
6966 AOT_hence ‹□x[F]› for F by (metis "en-eq:2[1]" "≡E"(1))
6967 AOT_hence ‹∀F □x[F]› by (rule GEN)
6968 AOT_hence ‹□∀F x[F]› by (rule BF[THEN "→E"])
6969 moreover AOT_have ‹□A!x› using x_abs
6970 using "oa-facts:2" "vdash-properties:10" by blast
6971 ultimately AOT_have r: ‹□(A!x & ∀F x[F])›
6972 by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
6973 AOT_show ‹□Universal(x)›
6974 by (AOT_subst "«Universal(x)»" "«A!x & ∀F x[F]»")
6975 (auto simp add: "df-null-uni:2" "≡Df" r)
6976qed
6977
6978AOT_theorem "null-uni-facts:3": ‹Null(a⇩∅)›
6979 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1"])
6980 apply (simp add: "null-uni-uniq:3")
6981 using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:3"]
6982 "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:1"[unvarify x, THEN RN, OF "null-uni-uniq:3"], THEN "≡E"(1)]
6983 by blast
6984
6985AOT_theorem "null-uni-facts:4": ‹Universal(a⇩V)›
6986 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2"])
6987 apply (simp add: "null-uni-uniq:4")
6988 using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:4"]
6989 "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:2"[unvarify x, THEN RN, OF "null-uni-uniq:4"], THEN "≡E"(1)]
6990 by blast
6991
6992AOT_theorem "null-uni-facts:5": ‹a⇩∅ ≠ a⇩V›
6993proof (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
6994 rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
6995 rule "≡⇩d⇩fI"[OF "=-infix"];
6996 rule "raa-cor:2")
6997 AOT_obtain x where nullx: ‹Null(x)›
6998 by (metis "instantiation" "df-null-uni-terms:1" "existential:1" "null-uni-facts:3"
6999 "null-uni-uniq:3" "rule-id-def:2:b[zero]")
7000 AOT_hence act_null: ‹❙𝒜Null(x)› by (metis "nec-imp-act" "null-uni-facts:1" "vdash-properties:10")
7001 AOT_assume ‹❙ιx Null(x) = ❙ιx Universal(x)›
7002 AOT_hence ‹❙𝒜∀x(Null(x) ≡ Universal(x))›
7003 using "actual-desc:5"[THEN "→E"] by blast
7004 AOT_hence ‹∀x ❙𝒜(Null(x) ≡ Universal(x))›
7005 by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
7006 AOT_hence ‹❙𝒜Null(x) ≡ ❙𝒜Universal(x)›
7007 using "Act-Basic:5" "≡E"(1) "rule-ui:3" by blast
7008 AOT_hence ‹❙𝒜Universal(x)› using act_null "≡E" by blast
7009 AOT_hence ‹Universal(x)› by (metis RN "≡E"(1) "null-uni-facts:2" "sc-eq-fur:2" "vdash-properties:10")
7010 AOT_hence ‹∀F x[F]› using "≡⇩d⇩fE"[OF "df-null-uni:2"] "&E" by metis
7011 moreover AOT_have ‹¬∃F x[F]› using nullx "≡⇩d⇩fE"[OF "df-null-uni:1"] "&E" by metis
7012 ultimately AOT_show ‹p & ¬p› for p by (metis "cqt-further:1" "raa-cor:3" "vdash-properties:10")
7013qed
7014
7015AOT_theorem "null-uni-facts:6": ‹a⇩∅ = ❙ιx(A!x & ∀F (x[F] ≡ F ≠ F))›
7016proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7017 AOT_show ‹❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7018 by (simp add: "A-descriptions")
7019next
7020 AOT_show ‹a⇩∅↓›
7021 by (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7022 (simp add: "null-uni-uniq:3")
7023next
7024 AOT_have ‹❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7025 by (simp add: "A-descriptions")
7026 AOT_hence 1: ‹❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F)) = ❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7027 using "rule=I:1" by blast
7028 AOT_show ‹[A!]a⇩∅ & [A!]❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7029 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"]; rule "&I")
7030 apply (meson "≡⇩d⇩fE" "Conjunction Simplification"(1) "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3" "null-uni-uniq:3" "rule-id-def:2:a[zero]" "vdash-properties:10")
7031 using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7032next
7033 AOT_show ‹∀F (a⇩∅[F] ≡ ❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F])›
7034 proof (rule GEN)
7035 fix F
7036 AOT_have ‹¬a⇩∅[F]›
7037 by (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7038 (metis (no_types, lifting) "≡⇩d⇩fE" "&E"(2) "∨I"(2) "∨E"(3)
7039 "df-null-uni:1" "df-null-uni-terms:1" "existential:2[const_var]" "null-uni-facts:3"
7040 "raa-cor:2" "rule-id-def:2:a[zero]" "russell-axiom[enc,1].ψ_denotes_asm")
7041 moreover AOT_have ‹¬❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7042 proof(rule "raa-cor:2")
7043 AOT_assume 0: ‹❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7044 AOT_hence ‹❙𝒜(F ≠ F)› using "desc-nec-encode"[THEN "≡E"(1), OF 0] by blast
7045 moreover AOT_have ‹¬❙𝒜(F ≠ F)›
7046 using "≡⇩d⇩fE" "id-act:2" "id-eq:1" "≡E"(2) "=-infix" "raa-cor:3" by blast
7047 ultimately AOT_show ‹❙𝒜(F ≠ F) & ¬❙𝒜(F ≠ F)› by (rule "&I")
7048 qed
7049 ultimately AOT_show ‹a⇩∅[F] ≡ ❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7050 using "deduction-theorem" "≡I" "raa-cor:4" by blast
7051 qed
7052qed
7053
7054AOT_theorem "null-uni-facts:7": ‹a⇩V = ❙ιx(A!x & ∀F (x[F] ≡ F = F))›
7055proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7056 AOT_show ‹❙ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7057 by (simp add: "A-descriptions")
7058next
7059 AOT_show ‹a⇩V↓›
7060 by (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7061 (simp add: "null-uni-uniq:4")
7062next
7063 AOT_have ‹❙ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7064 by (simp add: "A-descriptions")
7065 AOT_hence 1: ‹❙ιx([A!]x & ∀F (x[F] ≡ F = F)) = ❙ιx([A!]x & ∀F (x[F] ≡ F = F))›
7066 using "rule=I:1" by blast
7067 AOT_show ‹[A!]a⇩V & [A!]❙ιx([A!]x & ∀F (x[F] ≡ F = F))›
7068 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"]; rule "&I")
7069 apply (meson "≡⇩d⇩fE" "Conjunction Simplification"(1) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-def:2:a[zero]" "vdash-properties:10")
7070 using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7071next
7072 AOT_show ‹∀F (a⇩V[F] ≡ ❙ιx([A!]x & ∀F (x[F] ≡ F = F))[F])›
7073 proof (rule GEN)
7074 fix F
7075 AOT_have ‹a⇩V[F]›
7076 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7077 using "≡⇩d⇩fE" "&E"(2) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-def:2:a[zero]" "rule-ui:3" by blast
7078 moreover AOT_have ‹❙ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7079 using "RA[2]" "desc-nec-encode" "id-eq:1" "≡E"(2) by fastforce
7080 ultimately AOT_show ‹a⇩V[F] ≡ ❙ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7081 using "deduction-theorem" "≡I" by simp
7082 qed
7083qed
7084
7085AOT_theorem "aclassical:1": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])›
7086proof(rule GEN)
7087 fix R
7088 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]zy] & ¬y[F]))›
7089 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7090 AOT_have a_enc: ‹a[λz [R]za]›
7091 proof (rule "raa-cor:1")
7092 AOT_assume 0: ‹¬a[λz [R]za]›
7093 AOT_hence ‹¬∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7094 by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]za]»"],
7095 THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7096 "cqt:2[lambda]"
7097 AOT_hence ‹∀y ¬(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7098 using "cqt-further:4" "vdash-properties:10" by blast
7099 AOT_hence ‹¬(A!a & [λz [R]za] = [λz [R]za] & ¬a[λz [R]za])› using "∀E" by blast
7100 AOT_hence ‹(A!a & [λz [R]za] = [λz [R]za]) → a[λz [R]za]›
7101 by (metis "&I" "deduction-theorem" "raa-cor:3")
7102 moreover AOT_have ‹[λz [R]za] = [λz [R]za]›
7103 by (rule "=I") "cqt:2[lambda]"
7104 ultimately AOT_have ‹a[λz [R]za]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7105 AOT_thus ‹a[λz [R]za] & ¬a[λz [R]za]›
7106 using 0 "&I" by blast
7107 qed
7108 AOT_hence ‹∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7109 by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7110 then AOT_obtain b where b_prop: ‹A!b & [λz [R]za] = [λz [R]zb] & ¬b[λz [R]za]›
7111 using "∃E"[rotated] by blast
7112 AOT_have ‹a ≠ b›
7113 apply (rule "≡⇩d⇩fI"[OF "=-infix"])
7114 using a_enc b_prop[THEN "&E"(2)]
7115 using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7116 AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]za] = [λz [R]zb]›
7117 using b_prop "&E" a_prop "&I" by meson
7118 AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]za] = [λz [R]zy])› by (rule "∃I")
7119 AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])› by (rule "∃I")
7120qed
7121
7122AOT_theorem "aclassical:2": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])›
7123proof(rule GEN)
7124 fix R
7125 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]yz] & ¬y[F]))›
7126 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7127 AOT_have a_enc: ‹a[λz [R]az]›
7128 proof (rule "raa-cor:1")
7129 AOT_assume 0: ‹¬a[λz [R]az]›
7130 AOT_hence ‹¬∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7131 by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]az]»"],
7132 THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7133 "cqt:2[lambda]"
7134 AOT_hence ‹∀y ¬(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7135 using "cqt-further:4" "vdash-properties:10" by blast
7136 AOT_hence ‹¬(A!a & [λz [R]az] = [λz [R]az] & ¬a[λz [R]az])› using "∀E" by blast
7137 AOT_hence ‹(A!a & [λz [R]az] = [λz [R]az]) → a[λz [R]az]›
7138 by (metis "&I" "deduction-theorem" "raa-cor:3")
7139 moreover AOT_have ‹[λz [R]az] = [λz [R]az]›
7140 by (rule "=I") "cqt:2[lambda]"
7141 ultimately AOT_have ‹a[λz [R]az]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7142 AOT_thus ‹a[λz [R]az] & ¬a[λz [R]az]›
7143 using 0 "&I" by blast
7144 qed
7145 AOT_hence ‹∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7146 by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7147 then AOT_obtain b where b_prop: ‹A!b & [λz [R]az] = [λz [R]bz] & ¬b[λz [R]az]›
7148 using "∃E"[rotated] by blast
7149 AOT_have ‹a ≠ b›
7150 apply (rule "≡⇩d⇩fI"[OF "=-infix"])
7151 using a_enc b_prop[THEN "&E"(2)]
7152 using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7153 AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]az] = [λz [R]bz]›
7154 using b_prop "&E" a_prop "&I" by meson
7155 AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]az] = [λz [R]yz])› by (rule "∃I")
7156 AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])› by (rule "∃I")
7157qed
7158
7159AOT_theorem "aclassical:3": ‹∀F∃x∃y(A!x & A!y & x ≠ y & [λ [F]x] = [λ [F]y])›
7160proof(rule GEN)
7161 fix R
7162 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]y] & ¬y[F]))›
7163 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7164 AOT_have ‹[λz [R]a]↓› by "cqt:2[lambda]"
7165
7166 then AOT_obtain S where S_def: ‹S = [λz [R]a]›
7167 by (metis "instantiation" "rule=I:1" "existential:1" id_sym)
7168 AOT_have a_enc: ‹a[S]›
7169 proof (rule "raa-cor:1")
7170 AOT_assume 0: ‹¬a[S]›
7171 AOT_hence ‹¬∃y(A!y & S = [λz [R]y] & ¬y[S])›
7172 by (rule a_prop[THEN "&E"(2), THEN "∀E"(2)[where β=S],
7173 THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7174 AOT_hence ‹∀y ¬(A!y & S = [λz [R]y] & ¬y[S])›
7175 using "cqt-further:4" "vdash-properties:10" by blast
7176 AOT_hence ‹¬(A!a & S = [λz [R]a] & ¬a[S])› using "∀E" by blast
7177 AOT_hence ‹(A!a & S = [λz [R]a]) → a[S]›
7178 by (metis "&I" "deduction-theorem" "raa-cor:3")
7179 moreover AOT_have ‹S = [λz [R]a]› using S_def .
7180 ultimately AOT_have ‹a[S]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7181 AOT_thus ‹a[λz [R]a] & ¬a[λz [R]a]› by (metis "0" "raa-cor:3")
7182 qed
7183 AOT_hence ‹∃y(A!y & S = [λz [R]y] & ¬y[S])›
7184 by (rule a_prop[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1), rotated])
7185 then AOT_obtain b where b_prop: ‹A!b & S = [λz [R]b] & ¬b[S]›
7186 using "∃E"[rotated] by blast
7187 AOT_have 1: ‹a ≠ b›
7188 apply (rule "≡⇩d⇩fI"[OF "=-infix"])
7189 using a_enc b_prop[THEN "&E"(2)]
7190 using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7191 AOT_have a: ‹[λ [R]a] = ([R]a)›
7192 apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7193 by (meson "log-prop-prop:2")
7194 AOT_have b: ‹[λ [R]b] = ([R]b)›
7195 apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7196 by (meson "log-prop-prop:2")
7197 AOT_have ‹[λ [R]a] = [λ [R]b]›
7198 apply (rule "rule=E"[rotated, OF a[THEN id_sym]])
7199 apply (rule "rule=E"[rotated, OF b[THEN id_sym]])
7200 apply (rule "identity:4"[THEN "≡⇩d⇩fI", OF "&I", rotated])
7201 apply (rule "rule=E"[rotated, OF S_def])
7202 using b_prop "&E" apply blast
7203 apply (safe intro!: "&I")
7204 by (simp add: "log-prop-prop:2")+
7205 AOT_hence ‹A!a & A!b & a ≠ b & [λ [R]a] = [λ [R]b]›
7206 using 1 a_prop[THEN "&E"(1)] b_prop[THEN "&E"(1), THEN "&E"(1)] "&I" by auto
7207 AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λ [R]a] = [λ [R]y])› by (rule "∃I")
7208 AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λ [R]x] = [λ [R]y])› by (rule "∃I")
7209qed
7210
7211AOT_theorem aclassical2: ‹∃x∃y (A!x & A!y & x ≠ y & ∀F ([F]x ≡ [F]y))›
7212proof -
7213 AOT_have ‹∃x ∃y ([A!]x & [A!]y & x ≠ y &
7214 [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7215 by (rule "aclassical:1"[THEN "∀E"(1)[where τ="«[λxy ∀F ([F]x ≡ [F]y)]»"]])
7216 "cqt:2[lambda]"
7217 then AOT_obtain x where ‹∃y ([A!]x & [A!]y & x ≠ y &
7218 [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7219 using "∃E"[rotated] by blast
7220 then AOT_obtain y where 0: ‹([A!]x & [A!]y & x ≠ y &
7221 [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7222 using "∃E"[rotated] by blast
7223 AOT_have ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zx]x›
7224 apply (rule "β←C"(1))
7225 apply "cqt:2[lambda]"
7226 apply (fact "cqt:2[const_var]"[axiom_inst])
7227 apply (rule "β←C"(1))
7228 apply "cqt:2[lambda]"
7229 apply (simp add: "&I" "ex:1:a" prod_denotesI "rule-ui:3")
7230 by (simp add: "oth-class-taut:3:a" "universal-cor")
7231 AOT_hence ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zy]x›
7232 by (rule "rule=E"[rotated, OF 0[THEN "&E"(2)]])
7233 AOT_hence ‹[λxy ∀F ([F]x ≡ [F]y)]xy›
7234 by (rule "β→C"(1))
7235 AOT_hence ‹∀F ([F]x ≡ [F]y)›
7236 using "β→C"(1) old.prod.case by fast
7237 AOT_hence ‹[A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y)› using 0 "&E" "&I" by blast
7238 AOT_hence ‹∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I")
7239 AOT_thus ‹∃x∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I"(2))
7240qed
7241
7242AOT_theorem "kirchner-thm:1": ‹[λx φ{x}]↓ ≡ □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7243proof(rule "≡I"; rule "→I")
7244 AOT_assume ‹[λx φ{x}]↓›
7245 AOT_hence ‹□[λx φ{x}]↓› by (metis "exist-nec" "vdash-properties:10")
7246 moreover AOT_have ‹□[λx φ{x}]↓ → □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7247 proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7248 AOT_modally_strict {
7249 fix x y
7250 AOT_assume 0: ‹[λx φ{x}]↓›
7251 moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7252 ultimately AOT_have ‹[λx φ{x}]x ≡ [λx φ{x}]y›
7253 using "∀E" by blast
7254 AOT_thus ‹(φ{x} ≡ φ{y})›
7255 using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7256 }
7257 qed
7258 ultimately AOT_show ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7259 using "→E" by blast
7260next
7261 AOT_have ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y})) → □∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7262 proof(rule "RM:1"; rule "→I"; rule GEN)
7263 AOT_modally_strict {
7264 AOT_assume ‹∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7265 AOT_hence indisc: ‹φ{x} ≡ φ{y}› if ‹∀F([F]x ≡ [F]y)› for x y
7266 using "∀E"(2) "→E" that by blast
7267 AOT_show ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})› for y
7268 proof (rule "raa-cor:1")
7269 AOT_assume ‹¬(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7270 AOT_hence ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}) ∨ (¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7271 using "≡E"(1) "oth-class-taut:4:h" by blast
7272 moreover {
7273 AOT_assume 0: ‹∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}›
7274 AOT_obtain a where ‹∀F([F]a ≡ [F]y) & φ{a}›
7275 using "∃E"[rotated, OF 0[THEN "&E"(1)]] by blast
7276 AOT_hence ‹φ{y}› using indisc[THEN "≡E"(1)] "&E" by blast
7277 AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7278 }
7279 moreover {
7280 AOT_assume 0: ‹(¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7281 AOT_hence ‹∀x ¬(∀F([F]x ≡ [F]y) & φ{x})›
7282 using "&E"(1) "cqt-further:4" "→E" by blast
7283 AOT_hence ‹¬(∀F([F]y ≡ [F]y) & φ{y})› using "∀E" by blast
7284 AOT_hence ‹¬∀F([F]y ≡ [F]y) ∨ ¬φ{y}›
7285 using "≡E"(1) "oth-class-taut:5:c" by blast
7286 moreover AOT_have ‹∀F([F]y ≡ [F]y)› by (simp add: "oth-class-taut:3:a" "universal-cor")
7287 ultimately AOT_have ‹¬φ{y}› by (metis "¬¬I" "∨E"(2))
7288 AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7289 }
7290 ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7291 qed
7292 }
7293 qed
7294 moreover AOT_assume ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7295 ultimately AOT_have ‹□∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7296 using "→E" by blast
7297 AOT_thus ‹[λx φ{x}]↓›
7298 by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7299qed
7300
7301AOT_theorem "kirchner-thm:2": ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓ ≡ □∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7302proof(rule "≡I"; rule "→I")
7303 AOT_assume ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓›
7304 AOT_hence ‹□[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓› by (metis "exist-nec" "vdash-properties:10")
7305 moreover AOT_have ‹□[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓ → □∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7306 proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7307 AOT_modally_strict {
7308 fix x⇩1x⇩n y⇩1y⇩n :: ‹'a AOT_var›
7309 AOT_assume 0: ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓›
7310 moreover AOT_assume ‹∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)›
7311 ultimately AOT_have ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]x⇩1...x⇩n ≡ [λx⇩1...x⇩n φ{x⇩1...x⇩n}]y⇩1...y⇩n›
7312 using "∀E" by blast
7313 AOT_thus ‹(φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n})›
7314 using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7315 }
7316 qed
7317 ultimately AOT_show ‹□∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7318 using "→E" by blast
7319next
7320 AOT_have ‹□(∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))) →
7321 □∀y⇩1...∀y⇩n((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) ≡ φ{y⇩1...y⇩n})›
7322 proof(rule "RM:1"; rule "→I"; rule GEN)
7323 AOT_modally_strict {
7324 AOT_assume ‹∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7325 AOT_hence indisc: ‹φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}› if ‹∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)› for x⇩1x⇩n y⇩1y⇩n
7326 using "∀E"(2) "→E" that by blast
7327 AOT_show ‹(∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) ≡ φ{y⇩1...y⇩n}› for y⇩1y⇩n
7328 proof (rule "raa-cor:1")
7329 AOT_assume ‹¬((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) ≡ φ{y⇩1...y⇩n})›
7330 AOT_hence ‹((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) & ¬φ{y⇩1...y⇩n}) ∨
7331 (¬(∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) & φ{y⇩1...y⇩n})›
7332 using "≡E"(1) "oth-class-taut:4:h" by blast
7333 moreover {
7334 AOT_assume 0: ‹(∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) & ¬φ{y⇩1...y⇩n}›
7335 AOT_obtain a⇩1a⇩n where ‹∀F([F]a⇩1...a⇩n ≡ [F]y⇩1...y⇩n) & φ{a⇩1...a⇩n}›
7336 using "∃E"[rotated, OF 0[THEN "&E"(1)]] by blast
7337 AOT_hence ‹φ{y⇩1...y⇩n}› using indisc[THEN "≡E"(1)] "&E" by blast
7338 AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7339 }
7340 moreover {
7341 AOT_assume 0: ‹(¬((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n}))) & φ{y⇩1...y⇩n})›
7342 AOT_hence ‹∀x⇩1...∀x⇩n ¬(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})›
7343 using "&E"(1) "cqt-further:4" "→E" by blast
7344 AOT_hence ‹¬(∀F([F]y⇩1...y⇩n ≡ [F]y⇩1...y⇩n) & φ{y⇩1...y⇩n})› using "∀E" by blast
7345 AOT_hence ‹¬∀F([F]y⇩1...y⇩n ≡ [F]y⇩1...y⇩n) ∨ ¬φ{y⇩1...y⇩n}›
7346 using "≡E"(1) "oth-class-taut:5:c" by blast
7347 moreover AOT_have ‹∀F([F]y⇩1...y⇩n ≡ [F]y⇩1...y⇩n)›
7348 by (simp add: "oth-class-taut:3:a" "universal-cor")
7349 ultimately AOT_have ‹¬φ{y⇩1...y⇩n}› by (metis "¬¬I" "∨E"(2))
7350 AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7351 }
7352 ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7353 qed
7354 }
7355 qed
7356 moreover AOT_assume ‹□∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7357 ultimately AOT_have ‹□∀y⇩1...∀y⇩n((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) ≡ φ{y⇩1...y⇩n})›
7358 using "→E" by blast
7359 AOT_thus ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓›
7360 by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7361qed
7362
7363AOT_theorem "kirchner-thm-cor:1": ‹[λx φ{x}]↓ → ∀x∀y(∀F([F]x ≡ [F]y) → □(φ{x} ≡ φ{y}))›
7364proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7365 fix x y
7366 AOT_assume ‹[λx φ{x}]↓›
7367 AOT_hence ‹□∀x∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7368 by (rule "kirchner-thm:1"[THEN "≡E"(1)])
7369 AOT_hence ‹∀x□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7370 using CBF[THEN "→E"] by blast
7371 AOT_hence ‹□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7372 using "∀E" by blast
7373 AOT_hence ‹∀y □(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7374 using CBF[THEN "→E"] by blast
7375 AOT_hence ‹□(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7376 using "∀E" by blast
7377 AOT_hence ‹□∀F ([F]x ≡ [F]y) → □(φ{x} ≡ φ{y})›
7378 using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7379 moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7380 ultimately AOT_show ‹□(φ{x} ≡ φ{y})› using "→E" "ind-nec" by blast
7381qed
7382
7383AOT_theorem "kirchner-thm-cor:2":
7384 ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓ → ∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → □(φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7385proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7386 fix x⇩1x⇩n y⇩1y⇩n
7387 AOT_assume ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓›
7388 AOT_hence 0: ‹□∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n (∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7389 by (rule "kirchner-thm:2"[THEN "≡E"(1)])
7390 AOT_have ‹∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n □(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7391 proof(rule GEN; rule GEN)
7392 fix x⇩1x⇩n y⇩1y⇩n
7393 AOT_show ‹□(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7394 apply (rule "RM:1"[THEN "→E", rotated, OF 0]; rule "→I")
7395 using "∀E" by blast
7396 qed
7397 AOT_hence ‹∀y⇩1...∀y⇩n □(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7398 using "∀E" by blast
7399 AOT_hence ‹□(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7400 using "∀E" by blast
7401 AOT_hence ‹□(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7402 using "∀E" by blast
7403 AOT_hence 0: ‹□∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → □(φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n})›
7404 using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7405 moreover AOT_assume ‹∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)›
7406 moreover AOT_have ‹[λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]↓› by "cqt:2[lambda]"
7407 ultimately AOT_have ‹[λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]x⇩1...x⇩n ≡ [λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]y⇩1...y⇩n›
7408 using "∀E" by blast
7409 moreover AOT_have ‹[λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]y⇩1...y⇩n›
7410 apply (rule "β←C"(1))
7411 apply "cqt:2[lambda]"
7412 apply (fact "cqt:2[const_var]"[axiom_inst])
7413 by (simp add: RN GEN "oth-class-taut:3:a")
7414 ultimately AOT_have ‹[λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]x⇩1...x⇩n› using "≡E"(2) by blast
7415 AOT_hence ‹□∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)›
7416 using "β→C"(1) by blast
7417 AOT_thus ‹□(φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n})› using "→E" 0 by blast
7418qed
7419
7420AOT_define propositional :: ‹Π ⇒ φ› (‹Propositional'(_')›)
7421 "prop-prop1": ‹Propositional([F]) ≡⇩d⇩f ∃p(F = [λy p])›
7422
7423AOT_theorem "prop-prop2:1": ‹∀p [λy p]↓›
7424 by (rule GEN) "cqt:2[lambda]"
7425
7426AOT_theorem "prop-prop2:2": ‹[λν φ]↓›
7427 by "cqt:2[lambda]"
7428
7429AOT_theorem "prop-prop2:3": ‹F = [λy p] → □∀x([F]x ≡ p)›
7430proof (rule "→I")
7431 AOT_assume 0: ‹F = [λy p]›
7432 AOT_show ‹□∀x([F]x ≡ p)›
7433 by (rule "rule=E"[rotated, OF 0[symmetric]]; rule RN; rule GEN; rule "beta-C-meta"[THEN "→E"])
7434 "cqt:2[lambda]"
7435qed
7436
7437AOT_theorem "prop-prop2:4": ‹Propositional([F]) → □Propositional([F])›
7438proof(rule "→I")
7439 AOT_assume ‹Propositional([F])›
7440 AOT_hence ‹∃p(F = [λy p])› using "≡⇩d⇩fE"[OF "prop-prop1"] by blast
7441 then AOT_obtain p where ‹F = [λy p]› using "∃E"[rotated] by blast
7442 AOT_hence ‹□(F = [λy p])› using "id-nec:2" "modus-tollens:1" "raa-cor:3" by blast
7443 AOT_hence ‹∃p □(F = [λy p])› using "∃I" by fast
7444 AOT_hence 0: ‹□∃p (F = [λy p])› by (metis Buridan "vdash-properties:10")
7445 AOT_show ‹□Propositional([F])›
7446 apply (AOT_subst ‹«Propositional([F])»› ‹«∃p (F = [λy p])»›)
7447 using "prop-prop1" "≡Df" apply presburger
7448 by (fact 0)
7449qed
7450
7451AOT_define indicriminate :: ‹Π ⇒ φ› ("Indiscriminate'(_')")
7452 "prop-indis": ‹Indiscriminate([F]) ≡⇩d⇩f F↓ & □(∃x [F]x → ∀x [F]x)›
7453
7454AOT_theorem "prop-in-thm": ‹Propositional([Π]) → Indiscriminate([Π])›
7455proof(rule "→I")
7456 AOT_assume ‹Propositional([Π])›
7457 AOT_hence ‹∃p Π = [λy p]› using "≡⇩d⇩fE"[OF "prop-prop1"] by blast
7458 then AOT_obtain p where Π_def: ‹Π = [λy p]› using "∃E"[rotated] by blast
7459 AOT_show ‹Indiscriminate([Π])›
7460 proof (rule "≡⇩d⇩fI"[OF "prop-indis"]; rule "&I")
7461 AOT_show ‹Π↓›
7462 using Π_def by (meson "t=t-proper:1" "vdash-properties:6")
7463 next
7464 AOT_show ‹□(∃x [Π]x → ∀x [Π]x)›
7465 proof (rule "rule=E"[rotated, OF Π_def[symmetric]]; rule RN; rule "→I"; rule GEN)
7466 AOT_modally_strict {
7467 AOT_assume ‹∃x [λy p]x›
7468 then AOT_obtain a where ‹[λy p]a› using "∃E"[rotated] by blast
7469 AOT_hence 0: ‹p› by (metis "β→C"(1))
7470 AOT_show ‹[λy p]x› for x
7471 apply (rule "β←C"(1))
7472 apply "cqt:2[lambda]"
7473 apply (fact "cqt:2[const_var]"[axiom_inst])
7474 by (fact 0)
7475 }
7476 qed
7477 qed
7478qed
7479
7480AOT_theorem "prop-in-f:1": ‹Necessary([F]) → Indiscriminate([F])›
7481proof (rule "→I")
7482 AOT_assume ‹Necessary([F])›
7483 AOT_hence 0: ‹□∀x⇩1...∀x⇩n [F]x⇩1...x⇩n› using "≡⇩d⇩fE"[OF "contingent-properties:1"] by blast
7484 AOT_show ‹Indiscriminate([F])›
7485 by (rule "≡⇩d⇩fI"[OF "prop-indis"])
7486 (metis "0" "KBasic:1" "&I" "ex:1:a" "rule-ui:2[const_var]" "vdash-properties:6")
7487qed
7488
7489AOT_theorem "prop-in-f:2": ‹Impossible([F]) → Indiscriminate([F])›
7490proof (rule "→I")
7491 AOT_modally_strict {
7492 AOT_have ‹∀x ¬[F]x → (∃x [F]x → ∀x [F]x)›
7493 by (metis "instantiation" "cqt-orig:3" "Hypothetical Syllogism" "deduction-theorem" "raa-cor:3")
7494 }
7495 AOT_hence 0: ‹□∀x ¬[F]x → □(∃x [F]x → ∀x [F]x)›
7496 by (rule "RM:1")
7497 AOT_assume ‹Impossible([F])›
7498 AOT_hence ‹□∀x ¬[F]x› using "≡⇩d⇩fE"[OF "contingent-properties:2"] "&E" by blast
7499 AOT_hence 1: ‹□(∃x [F]x → ∀x [F]x)› using 0 "→E" by blast
7500 AOT_show ‹Indiscriminate([F])›
7501 by (rule "≡⇩d⇩fI"[OF "prop-indis"]; rule "&I")
7502 (simp add: "ex:1:a" "rule-ui:2[const_var]" 1)+
7503qed
7504
7505AOT_theorem "prop-in-f:3:a": ‹¬Indiscriminate([E!])›
7506proof(rule "raa-cor:2")
7507 AOT_assume ‹Indiscriminate([E!])›
7508 AOT_hence 0: ‹□(∃x [E!]x → ∀x [E!]x)›
7509 using "≡⇩d⇩fE"[OF "prop-indis"] "&E" by blast
7510 AOT_hence ‹◇∃x [E!]x → ◇∀x [E!]x›
7511 using "KBasic:13" "vdash-properties:10" by blast
7512 moreover AOT_have ‹◇∃x [E!]x›
7513 by (simp add: "thm-cont-e:3")
7514 ultimately AOT_have ‹◇∀x [E!]x›
7515 by (metis "vdash-properties:6")
7516 AOT_thus ‹p & ¬p› for p
7517 by (metis "≡⇩d⇩fE" "conventions:5" "o-objects-exist:5" "reductio-aa:1")
7518qed
7519
7520AOT_theorem "prop-in-f:3:b": ‹¬Indiscriminate([E!]⇧-)›
7521proof (rule "rule=E"[rotated, OF "rel-neg-T:2"[symmetric]]; rule "raa-cor:2")
7522 AOT_assume ‹Indiscriminate([λx ¬[E!]x])›
7523 AOT_hence 0: ‹□(∃x [λx ¬[E!]x]x → ∀x [λx ¬[E!]x]x)›
7524 using "≡⇩d⇩fE"[OF "prop-indis"] "&E" by blast
7525 AOT_hence ‹□∃x [λx ¬[E!]x]x → □∀x [λx ¬[E!]x]x›
7526 using "→E" "qml:1" "vdash-properties:1[2]" by blast
7527 moreover AOT_have ‹□∃x [λx ¬[E!]x]x›
7528 apply (AOT_subst ‹λκ. «[λx ¬[E!]x]κ»› ‹λκ. «¬[E!]κ»›)
7529 apply (rule "beta-C-meta"[THEN "→E"])
7530 apply "cqt:2[lambda]"
7531 by (metis (full_types) "B◇" RN "T◇" "cqt-further:2" "o-objects-exist:5" "vdash-properties:10")
7532 ultimately AOT_have 1: ‹□∀x [λx ¬[E!]x]x›
7533 by (metis "vdash-properties:6")
7534 AOT_have ‹□∀x ¬[E!]x›
7535 apply (AOT_subst_rev ‹λκ. «[λx ¬[E!]x]κ»› ‹λκ. «¬[E!]κ»›)
7536 apply (rule "beta-C-meta"[THEN "→E"])
7537 apply "cqt:2[lambda]"
7538 by (fact 1)
7539 AOT_hence ‹∀x □¬[E!]x› by (metis "CBF" "vdash-properties:10")
7540 moreover AOT_obtain a where abs_a: ‹O!a›
7541 using "instantiation" "o-objects-exist:1" "qml:2" "vdash-properties:1[2]" "vdash-properties:6" by blast
7542 ultimately AOT_have ‹□¬[E!]a› using "∀E" by blast
7543 AOT_hence 2: ‹¬◇[E!]a› by (metis "≡⇩d⇩fE" "conventions:5" "reductio-aa:1")
7544 AOT_have ‹A!a›
7545 apply (rule "=⇩d⇩fI"(2)[OF AOT_abstract])
7546 apply "cqt:2[lambda]"
7547 apply (rule "β←C"(1))
7548 apply "cqt:2[lambda]"
7549 using "cqt:2[const_var]"[axiom_inst] apply blast
7550 by (fact 2)
7551 AOT_thus ‹p & ¬p› for p using abs_a
7552 by (metis "≡E"(1) "oa-contingent:2" "reductio-aa:1")
7553qed
7554
7555AOT_theorem "prop-in-f:3:c": ‹¬Indiscriminate(O!)›
7556proof(rule "raa-cor:2")
7557 AOT_assume ‹Indiscriminate(O!)›
7558 AOT_hence 0: ‹□(∃x O!x → ∀x O!x)›
7559 using "≡⇩d⇩fE"[OF "prop-indis"] "&E" by blast
7560 AOT_hence ‹□∃x O!x → □∀x O!x›
7561 using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7562 moreover AOT_have ‹□∃x O!x›
7563 using "o-objects-exist:1" by blast
7564 ultimately AOT_have ‹□∀x O!x›
7565 by (metis "vdash-properties:6")
7566 AOT_thus ‹p & ¬p› for p
7567 by (metis "o-objects-exist:3" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7568qed
7569
7570AOT_theorem "prop-in-f:3:d": ‹¬Indiscriminate(A!)›
7571proof(rule "raa-cor:2")
7572 AOT_assume ‹Indiscriminate(A!)›
7573 AOT_hence 0: ‹□(∃x A!x → ∀x A!x)›
7574 using "≡⇩d⇩fE"[OF "prop-indis"] "&E" by blast
7575 AOT_hence ‹□∃x A!x → □∀x A!x›
7576 using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7577 moreover AOT_have ‹□∃x A!x›
7578 using "o-objects-exist:2" by blast
7579 ultimately AOT_have ‹□∀x A!x›
7580 by (metis "vdash-properties:6")
7581 AOT_thus ‹p & ¬p› for p
7582 by (metis "o-objects-exist:4" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7583qed
7584
7585AOT_theorem "prop-in-f:4:a": ‹¬Propositional(E!)›
7586 using "modus-tollens:1" "prop-in-f:3:a" "prop-in-thm" by blast
7587
7588AOT_theorem "prop-in-f:4:b": ‹¬Propositional(E!⇧-)›
7589 using "modus-tollens:1" "prop-in-f:3:b" "prop-in-thm" by blast
7590
7591AOT_theorem "prop-in-f:4:c": ‹¬Propositional(O!)›
7592 using "modus-tollens:1" "prop-in-f:3:c" "prop-in-thm" by blast
7593
7594AOT_theorem "prop-in-f:4:d": ‹¬Propositional(A!)›
7595 using "modus-tollens:1" "prop-in-f:3:d" "prop-in-thm" by blast
7596
7597AOT_theorem "prop-prop-nec:1": ‹◇∃p (F = [λy p]) → ∃p(F = [λy p])›
7598proof(rule "→I")
7599 AOT_assume ‹◇∃p (F = [λy p])›
7600 AOT_hence ‹∃p ◇(F = [λy p])›
7601 by (metis "BF◇" "vdash-properties:10")
7602 then AOT_obtain p where ‹◇(F = [λy p])› using "∃E"[rotated] by blast
7603 AOT_hence ‹F = [λy p]› by (metis "derived-S5-rules:2" emptyE "id-nec:2" "vdash-properties:6")
7604 AOT_thus ‹∃p(F = [λy p])› by (rule "∃I")
7605qed
7606
7607AOT_theorem "prop-prop-nec:2": ‹∀p (F ≠ [λy p]) → □∀p(F ≠ [λy p])›
7608proof(rule "→I")
7609 AOT_assume ‹∀p (F ≠ [λy p])›
7610 AOT_hence ‹(F ≠ [λy p])› for p
7611 using "∀E" by blast
7612 AOT_hence ‹□(F ≠ [λy p])› for p
7613 by (rule "id-nec2:2"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7614 AOT_hence ‹∀p □(F ≠ [λy p])› by (rule GEN)
7615 AOT_thus ‹□∀p (F ≠ [λy p])› using BF[THEN "→E"] by fast
7616qed
7617
7618AOT_theorem "prop-prop-nec:3": ‹∃p (F = [λy p]) → □∃p(F = [λy p])›
7619proof(rule "→I")
7620 AOT_assume ‹∃p (F = [λy p])›
7621 then AOT_obtain p where ‹(F = [λy p])› using "∃E"[rotated] by blast
7622 AOT_hence ‹□(F = [λy p])› by (metis "id-nec:2" "vdash-properties:6")
7623 AOT_hence ‹∃p□(F = [λy p])› by (rule "∃I")
7624 AOT_thus ‹□∃p(F = [λy p])› by (metis Buridan "vdash-properties:10")
7625qed
7626
7627AOT_theorem "prop-prop-nec:4": ‹◇∀p (F ≠ [λy p]) → ∀p(F ≠ [λy p])›
7628proof(rule "→I")
7629 AOT_assume ‹◇∀p (F ≠ [λy p])›
7630 AOT_hence ‹∀p ◇(F ≠ [λy p])› by (metis "Buridan◇" "vdash-properties:10")
7631 AOT_hence ‹◇(F ≠ [λy p])› for p
7632 using "∀E" by blast
7633 AOT_hence ‹F ≠ [λy p]› for p
7634 by (rule "id-nec2:3"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7635 AOT_thus ‹∀p (F ≠ [λy p])› by (rule GEN)
7636qed
7637
7638AOT_theorem "enc-prop-nec:1": ‹◇∀F (x[F] → ∃p(F = [λy p])) → ∀F(x[F] → ∃p (F = [λy p]))›
7639proof(rule "→I"; rule GEN; rule "→I")
7640 fix F
7641 AOT_assume ‹◇∀F (x[F] → ∃p(F = [λy p]))›
7642 AOT_hence ‹∀F ◇(x[F] → ∃p(F = [λy p]))›
7643 using "Buridan◇" "vdash-properties:10" by blast
7644 AOT_hence 0: ‹◇(x[F] → ∃p(F = [λy p]))› using "∀E" by blast
7645 AOT_assume ‹x[F]›
7646 AOT_hence ‹□x[F]› by (metis "en-eq:2[1]" "≡E"(1))
7647 AOT_hence ‹◇∃p(F = [λy p])›
7648 using 0 by (metis "KBasic2:4" "≡E"(1) "vdash-properties:10")
7649 AOT_thus ‹∃p(F = [λy p])›
7650 using "prop-prop-nec:1"[THEN "→E"] by blast
7651qed
7652
7653AOT_theorem "enc-prop-nec:2": ‹∀F (x[F] → ∃p(F = [λy p])) → □∀F(x[F] → ∃p (F = [λy p]))›
7654 using "derived-S5-rules:1"[where Γ="{}", simplified, OF "enc-prop-nec:1"]
7655 by blast
7656
7657
7658end
7659